Optimal and typical L2 discrepancy of 2-dimensional lattices

TL;DR

This paper analyzes the L2 discrepancy of 2D lattices, focusing on optimal configurations and typical behavior for both rational and irrational cases.

Contribution

The paper provides a full characterization of optimal lattices using continued fractions and derives asymptotics for specific irrational numbers.

Findings

Optimal L2 discrepancy is characterized via continued fraction partial quotients.

Asymptotics for L2 discrepancy are computed for quadratic irrationals and Euler’s number e.

Limit distributions are derived for randomly chosen rational and irrational lattices.

Abstract

We undertake a detailed study of the \documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}L2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal \documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}L2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler’s number e. In the metric theory, we find the asymptotics of the \documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}L2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.

Full text

Introduction

The extreme discrepancy of a finite point set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \subset [0,1)^d$$\end{document} in the unit cube is defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_{\textrm{extr}} (P) = \sup _{A \subseteq [0,1)^d} \left| |P \cap A| - |P| \lambda (A) \right| , \end{aligned}$$\end{document}

where the supremum is over all axis-parallel boxes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=[a_1, b_1) \times \cdots \times [a_d, b_d) \subseteq [0,1)^d$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is the Lebesgue measure. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy is defined as the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} average over all axis-parallel boxes anchored at the origin:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (P) = \left( \int _{[0,1]^d} \left( |P \cap [0,x_1) \times \cdots \times [0,x_d) | -|P|x_1 \cdots x_d \right) ^2 \, \textrm{d} x_1 \cdots \textrm{d} x_d \right) ^{1/2}. \end{aligned}$$\end{document}

Clearly, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2 (P) \le D_{\textrm{extr}}(P)$$\end{document} . The extreme and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy are common measures of equidistribution, with direct applications to numerical integration; for a general introduction we refer to the monograph Drmota–Tichy [16]. In dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} , a seminal result of K. Roth [26] states that every finite point set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \subset [0,1)^2$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(P) \gg \sqrt{\log |P|}$$\end{document} with a universal implied constant. This is known to be sharp, with several explicit constructions e.g. based on digital nets attaining the optimal order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(P) \ll \sqrt{\log |P|}$$\end{document} , see [15]. The corresponding result for the extreme discrepancy due to W. Schmidt [16, p. 44] states that for every finite point set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \subset [0,1)^2$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}} (P) \gg \log |P|$$\end{document} with a universal implied constant, and this is again sharp with several explicit constructions attaining the optimal order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}} (P) \ll \log |P|$$\end{document} . In contrast, in dimensions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 3$$\end{document} the best known constructions of finite point sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \subset [0,1)^d$$\end{document} satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}} (P) \ll (\log |P|)^{d-1}$$\end{document} , but the optimal order remains an important open problem.

Given a positive integer N and integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1, \ldots , p_d$$\end{document} , the set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{p_1, \ldots , p_d,N} = \left\{ \left( \left\{ \frac{np_1}{N} \right\} , \ldots , \left\{ \frac{np_d}{N} \right\} \right) \in [0,1)^d \,: \, 0 \le n \le N-1 \right\} , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \cdot \}$$\end{document} denotes the fractional part function, is called a rank-1 lattice rule or a Korobov lattice. The terminology is explained by the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{p_1,\ldots , p_d,N}$$\end{document} is the intersection of the unit cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1)^d$$\end{document} and the dual of the lattice1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma = \{ (n_1,\ldots , n_d) \in \mathbb {Z}^d \,: \, n_1 p_1 + \cdots + n_d p_d \equiv 0 \pmod {N} \}$$\end{document} . Korobov lattices have been extensively used as quadrature rules in numerical integration. Finding lattice points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p_1,\ldots , p_d) \in \mathbb {Z}^d$$\end{document} that make the approximation error

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{[0,1]^d} f(x) \, \textrm{d}x \approx \frac{1}{N} \sum _{n=0}^{N-1} f \left( \left\{ \frac{np_1}{N} \right\} , \ldots , \left\{ \frac{np_d}{N} \right\} \right) \end{aligned}$$\end{document}

suitably small for given classes of integrable functions f is known as the method of “good lattice points”. We refer to the monograph of Dick, Kritzer and Pillichshammer [14] for a comprehensive survey and a history of lattice rules.

Bykovskii [11] showed that in any dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 3$$\end{document} , for any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document} there exist integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1, \ldots , p_d$$\end{document} , each coprime with N, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}}(\mathcal {L}_{p_1, \ldots , p_d,N}) \ll (\log N)^{d-1} \log \log N$$\end{document} . Note that this matches the best known constructions up to a factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log \log N$$\end{document} . The proof is probabilistic, see also [19, 20]. We refer to [14, Chapter 5] for an algorithm that produces a good lattice point such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}}(\mathcal {L}_{p_1, \ldots , p_d,N}) \ll (\log N)^d$$\end{document} , and for results on the discrepancy with respect to all convex sets.

Dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} is rather special. In this case, given a positive integer N, and integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1, p_2$$\end{document} , both coprime with N, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{p_1,p_2,N} = \mathcal {L}_{p,1,N}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=p_2^* p_1$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_2^*$$\end{document} is the multiplicative inverse of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_2 \pmod {N}$$\end{document} . In particular, Korobov lattices are parametrized by a single rational number p/N, and their distribution properties can be characterized in terms of the continued fraction expansion of p/N. The main goal of this paper is to study the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy of 2-dimensional Korobov lattices, and their analogues corresponding to irrational values of the parameter.

More precisely, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \mathbb {R}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \in \mathbb {N}$$\end{document} , we will consider the N-element set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(\alpha , N) = \left\{ \left( \{ n \alpha \}, \frac{n}{N} \right) \in [0,1)^2 \,: \, 0 \le n \le N-1 \right\} , \end{aligned}$$\end{document}

and the 2N-element set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S(\alpha , N) = \left\{ \left( \{ \pm n \alpha \}, \frac{n}{N} \right) \in [0,1)^2 \,: \, 0 \le n \le N-1 \right\} . \end{aligned}$$\end{document}

Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} is the intersection of the unit square \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1)^2$$\end{document} and the lattice spanned by the vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , 1/N)$$\end{document} and (1, 0). We call \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} the symmetrization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} ; more precisely, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} is the union of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} and its reflection about the vertical line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1/2$$\end{document} . We study both rational and irrational values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} .

The equidistribution properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} , in particular their \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy, are closely related to the Diophantine approximation properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} . Throughout this paper, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =[a_0;a_1,a_2,\dots ]$$\end{document} will denote the (finite or infinite) continued fraction expansion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_k/q_k=[a_0;a_1,\dots , a_k]$$\end{document} its convergents. In the rational case it will not matter which of the two possible expansions is chosen. Roughly speaking, we will show that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \approx q_K$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (S(\alpha , N)) \approx \sum _{k=1}^K a_k^2 \quad \text {and} \quad D_2^2(L(\alpha , N)) \approx \sum _{k=1}^K a_k^2 + \left( \sum _{k=1}^K (-1)^k a_k \right) ^2. \end{aligned}$$\end{document}

See Propositions 7 and 8 below for a precise formulation.

Our first result characterizes all irrationals for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , q_K)$$\end{document} resp. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , q_K)$$\end{document} attains optimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \rightarrow \infty $$\end{document} . We also consider the same problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document} . The first equivalence below generalizes a result of Davenport [12], who showed that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} attains optimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is badly approximable, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \ll 1$$\end{document} .

Theorem 1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =[a_0;a_1,a_2, \dots ]$$\end{document} be irrational. We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} D_2 (S(\alpha ,N)) \ll \sqrt{\log N} \,\,&\Longleftrightarrow \,\, D_2 (S(\alpha ,q_K)) \ll \sqrt{\log q_K} \,\, \Longleftrightarrow \,\, \frac{1}{K} \sum _{k=1}^K a_k^2 \ll 1, \\ D_2 (L(\alpha ,q_K)) \ll \sqrt{\log q_K} \,\,&\Longleftrightarrow \,\, \frac{1}{K} \sum _{k=1}^K a_k^2 \ll 1 \text { and } \frac{1}{\sqrt{K}} \left| \sum _{k=1}^K (-1)^k a_k \right| \ll 1. \end{aligned} \end{aligned}$$\end{document}

Remark 1

We also give an almost complete answer for the unsymmetrized lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} with general N: under the assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \ll \sqrt{k}/(\log k)^2$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (L(\alpha ,N)) \ll \sqrt{\log N} \,\, \Longleftrightarrow \,\, \frac{1}{K} \sum _{k=1}^K a_k^2 \ll 1 \text { and } \frac{1}{\sqrt{K}} \left| \sum _{k=1}^K (-1)^k a_k \right| \ll 1. \end{aligned}$$\end{document}

In the special case of a badly approximable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , this equivalence was observed in [5, 7]. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{-1} \sum _{k=1}^K a_k^2 \ll 1$$\end{document} implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \ll \sqrt{k}$$\end{document} ; we do not know whether the slightly stronger extra assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \ll \sqrt{k}/(\log k)^2$$\end{document} can be removed.

More precise results can be deduced for an irrational \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} whose continued fraction expansion is explicitly known. The most interesting case is that of quadratic irrationals, whose continued fractions are of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =[a_0;a_1,\dots , a_r,\overline{a_{r+1}, \dots , a_{r+p}}]$$\end{document} , where the overline denotes the period. Note that in this case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K (-1)^k a_k = A(\alpha ) K +O(1)$$\end{document} with some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\alpha )$$\end{document} . In fact, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\alpha )=0$$\end{document} if p is odd, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\alpha )=p^{-1} \sum _{k=1}^p (-1)^{r+k} a_{r+k}$$\end{document} (possibly zero) if p is even. We also have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log q_K=\Lambda (\alpha ) K+O(1)$$\end{document} with some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda (\alpha )>0$$\end{document} . In fact, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda (\alpha ) = p^{-1} \log \eta $$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta >1$$\end{document} is the larger of the two eigenvalues of the matrix

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} a_{r+1} \end{array} \right) \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} a_{r+2} \end{array} \right) \cdots \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} a_{r+p} \end{array} \right) . \end{aligned}$$\end{document}

Theorem 2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} be a quadratic irrational, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\alpha )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda (\alpha )$$\end{document} be as above. There exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(\alpha )>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2(S(\alpha , N)) = c(\alpha ) \log N +O(1), \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2(L(\alpha , N)) = \left\{ \begin{array}{ll} \frac{3}{2} c(\alpha ) \log N + O((\log \log N)^4) &{} \text {if } A(\alpha )=0, \\ \frac{A(\alpha )^2}{144 \Lambda (\alpha )^2} (\log N)^2 + O(\log N) &{} \text {if } A(\alpha ) \ne 0. \end{array} \right. \end{aligned}$$\end{document}

The implied constants depend only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} .

We proved the same result for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} with the slightly worse error term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log \log N)$$\end{document} in a previous paper [9]. In contrast to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\alpha )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda (\alpha )$$\end{document} , there seems to be no simple way to compute the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(\alpha )$$\end{document} directly from the continued fraction expansion. The latter constant first appeared in certain lattice point counting problems studied in detail by Beck [1–3], who showed that it is related to the arithmetic of the ring of algebraic integers of the real quadratic field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}(\alpha )$$\end{document} , and computed its explicit value for any quadratic irrational; for instance,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c \left( \frac{1+\sqrt{5}}{2} \right) = \frac{1}{30 \sqrt{5} \log \frac{1+\sqrt{5}}{2}} \quad \text {and} \quad c(\sqrt{3}) = \frac{1}{12 \sqrt{3} \log (2+\sqrt{3})}. \end{aligned}$$\end{document}

Precise results also follow for non-badly approximable irrationals whose continued fraction expansions are explicitly known. Consider Euler’s number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e=[2;1,2,1,1,4,1,\dots , 1,2n,1,\dots ]$$\end{document} as an illustration. Since the “period length” is odd, the square of the alternating sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sum _{k=1}^K (-1)^k a_k)^2 \ll K^2$$\end{document} is negligible compared to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K a_k^2 = (4/81)K^3+O(K^2)$$\end{document} . Thus from our general results it easily follows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (S(e,N)) = \frac{1}{3\sqrt{30}} \left( \frac{\log N}{\log \log N} \right) ^{3/2} \left( 1 + O \left( \frac{\log \log \log N}{\log \log N} \right) \right) , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (L(e,N)) = \frac{1}{6\sqrt{5}} \left( \frac{\log N}{\log \log N} \right) ^{3/2} \left( 1 + O \left( \frac{\log \log \log N}{\log \log N} \right) \right) . \end{aligned}$$\end{document}

In contrast, e.g. for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tan 1 = [1;1,1,3,1,5,1,\dots , 2n-1,1, \dots ]$$\end{document} , the “period length” is even, and the alternating sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sum _{k=1}^K (-1)^k a_k)^2=K^4/16+O(K^3)$$\end{document} dominates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K a_k^2 = K^3/6 + O(K^2)$$\end{document} . Consequently,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (S(\tan 1,N)) = \frac{1}{3\sqrt{30}} \left( \frac{\log N}{\log \log N} \right) ^{3/2} \left( 1 + O \left( \frac{\log \log \log N}{\log \log N} \right) \right) , \end{aligned}$$\end{document}

but for the unsymmetrized lattice we have the larger order of magnitude

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (L(\tan 1, N)) = \frac{1}{12} \left( \frac{\log N}{\log \log N} \right) ^2 \left( 1 + O \left( \frac{\log \log \log N}{\log \log N} \right) \right) . \end{aligned}$$\end{document}

We also establish precise results for randomly chosen \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , starting with the asymptotics a.e. in the sense of the Lebesgue measure.

Theorem 3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} be a positive nondecreasing function on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\infty )$$\end{document} .

• (i)If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty } 1/\varphi (n) < \infty $$\end{document} , then for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} D_2(S(\alpha , N))&\le \varphi (\log N) + O(\log N \log \log N), \\ D_2(L(\alpha , N))&\le \varphi (\log N) + O(\log N \log \log N) \end{aligned} \end{aligned}$$\end{document}

with implied constants depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} .

• (ii)If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty } 1/\varphi (n) = \infty $$\end{document} , then for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2 (S(\alpha , N)) \ge \varphi (\log N) \quad \text {and} \quad D_2 (L(\alpha , N)) \ge \varphi (\log N) \quad \text {for infinitely many } N. \end{aligned}$$\end{document}

In particular, for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2 (S(\alpha , N)) \ll \log N (\log \log N)^{1+\varepsilon }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2 (L(\alpha , N)) \ll \log N (\log \log N)^{1+\varepsilon }$$\end{document} with any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , but these fail with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0$$\end{document} .

Our next result is the distributional analogue of Theorem 3, stating that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is chosen randomly from [0, 1] with an absolutely continuous distribution, then after suitable normalization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(S(\alpha , N))$$\end{document} converges to the standard Lévy distribution. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is chosen randomly with the Lebesgue measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} or the Gauss measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu (B)=(1/\log 2) \int _B 1/(1+x) \, \textrm{d}x$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B \subseteq [0,1]$$\end{document} Borel) as distribution, then we also estimate the rate of convergence in the Kolmogorov metric.

Theorem 4

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is a Borel probability measure on [0, 1] which is absolutely continuous with respect to the Lebesgue measure, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu \left( \left\{ \alpha \in [0,1] \,: \, 5 \pi ^3 \frac{D_2^2 (S(\alpha , N))}{(\log N)^2} \le t \right\} \right) \rightarrow \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x \qquad \text {as } N \rightarrow \infty . \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is either the Lebesgue measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} or the Gauss measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \ge 0} \left| \mu \left( \left\{ \alpha \in [0,1] \,: \, 5 \pi ^3 \frac{D_2^2 (S(\alpha , N))}{(\log N)^2} \le t \right\} \right) - \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x \right| \ll \frac{(\log \log N)^{1/3}}{(\log N)^{1/3}} \end{aligned}$$\end{document}

with a universal implied constant.

We conjecture that a similar result holds for the unsymmetrized lattice as well, i.e. if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is chosen randomly from [0, 1] with an absolutely continuous distribution, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(L(\alpha , N))/(\log N)^2$$\end{document} has a nondegenerate limit distribution as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document} .

Our results, especially Theorems 1, 3 and 4 should be compared to the corresponding properties of the extreme discrepancy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr},N} (n \alpha ) :=D_{\textrm{extr}} (\{ \{n \alpha \} \,: \, 1 \le n \le N \})$$\end{document} of the classical Kronecker sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ n \alpha \}$$\end{document} . Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{1 \le \ell \le N} D_{\textrm{extr},\ell } (n \alpha )$$\end{document} is, up to a factor of 2, equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}}(L(\alpha , N))$$\end{document} . Roughly speaking, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \approx q_K$$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{1 \le \ell \le N} D_{\textrm{extr},\ell }( n \alpha ) \approx \sum _{k=1}^K a_k$$\end{document} . We can characterize all irrationals for which the optimal rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log N$$\end{document} is attained as [16, p. 53]

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_{\textrm{extr},N} (n \alpha ) \ll \log N \,\, \Longleftrightarrow \,\, \frac{1}{K} \sum _{k=1}^K a_k \ll 1. \end{aligned}$$\end{document}

The extreme discrepancy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr},N}(n \alpha )$$\end{document} is also known to satisfy the same asymptotics a.e. as in Theorem 3 [16, p. 63]. A fortiori, the previous two results apply also to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{1 \le \ell \le N} D_{\textrm{extr},\ell }(n \alpha )$$\end{document} , and hence to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}}(L(\alpha , N))$$\end{document} . We mention two distributional analogues due to Kesten [22]:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \frac{D_{\textrm{extr},N}(n \alpha )}{\log N \log \log N}&\rightarrow \frac{2}{\pi ^2} \quad \text {in measure,} \\ \frac{\max _{1 \le \ell \le N}D_{\textrm{extr},\ell } (n \alpha )}{\log N \log \log N}&\rightarrow \frac{3}{\pi ^2} \quad \text {in measure.} \end{aligned} \end{aligned}$$\end{document}

Full limit laws for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr},N}(n \alpha )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{1 \le \ell \le N} D_{\textrm{extr},\ell } (n \alpha )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\textrm{extr}}(L(\alpha ,N))$$\end{document} remain challenging open problems.

As a curious observation, we mention that there exists an irrational \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \log N \ll D_2(S(\alpha , N)) \le D_{\textrm{extr}} (S(\alpha , N)) \ll \log N, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \log N \ll D_2(L(\alpha , N)) \le D_{\textrm{extr}} (L(\alpha , N)) \ll \log N, \end{aligned}$$\end{document}

i.e. both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} have optimal extreme discrepancy, but neither has optimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy. Indeed, it is easy to construct a sequence of positive integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{-1} \sum _{k=1}^K a_k \ll 1$$\end{document} but \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K a_k^2 \gg K^2$$\end{document} (e.g. let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k=k$$\end{document} if k is a power of 2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k=1$$\end{document} otherwise).

Consider now the case of a rational \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} . For the sake of simplicity, we will always assume that N is the denominator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} . That is, given a reduced fraction p/q, we study the q-element set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(p/q,q) = \left\{ \left( \left\{ \frac{np}{q} \right\} , \frac{n}{q} \right) \in [0,1)^2 \,: \, 0 \le n \le q-1 \right\} , \end{aligned}$$\end{document}

and the 2q-element set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S(p/q,q) = \left\{ \left( \left\{ \pm \frac{np}{q} \right\} , \frac{n}{q} \right) \in [0,1)^2 \,: \, 0 \le n \le q-1 \right\} . \end{aligned}$$\end{document}

Note that L(p/q, q) is the 2-dimensional Korobov lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{p,1,q}$$\end{document} . The characterization of all sets of rationals for which the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy is optimal is exactly the same as in the irrational case.

Theorem 5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R} \subseteq \mathbb {Q}$$\end{document} be an arbitrary set of reduced fractions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/q=[a_0;a_1,\dots , a_r]$$\end{document} . We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sup _{p/q \in \mathcal {R}} \frac{D_2(S(p/q,q))}{\sqrt{\log q}}<\infty \,\, \Longleftrightarrow \,\,&\sup _{p/q \in \mathcal {R}} \frac{1}{r} \sum _{k=1}^r a_k^2< \infty , \\ \sup _{p/q \in \mathcal {R}} \frac{D_2(L(p/q,q))}{\sqrt{\log q}}< \infty \,\, \Longleftrightarrow \,\,&\sup _{p/q \in \mathcal {R}} \frac{1}{r} \sum _{k=1}^r a_k^2< \infty \\&\quad \text { and } \sup _{p/q \in \mathcal {R}} \frac{1}{\sqrt{r}} \left| \sum _{k=1}^r (-1)^k a_k \right| < \infty . \end{aligned} \end{aligned}$$\end{document}

As an analogue of the metric results on typical values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} in the sense of the Lebesgue measure above, we also study the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy for typical values of rationals. In this case, “typical” means choosing p/q randomly from the set of Farey fractions, that is, the set of all reduced rationals with bounded denominator.

Theorem 6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_Q$$\end{document} denote the set of all reduced fractions in the interval (0, 1) with denominator at most Q. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \ge 2$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \ge 0} \left| \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, 5 \pi ^3 \frac{D_2^2 (S(p/q,q))}{(\log q)^2} \le t \right\} \right| - \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x \right| \ll \frac{1}{(\log Q)^{1/2}} \end{aligned}$$\end{document}

with a universal implied constant.

We conjecture that a similar result holds for the unsymmetrized lattice as well, i.e. if p/q is chosen randomly from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_Q$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(L(p/q,q))/(\log q)^2$$\end{document} has a nondegenerate limit distribution as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \rightarrow \infty $$\end{document} .

In Sect. 2, we derive an explicit formula for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(S(\alpha , N))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(L(\alpha , N))$$\end{document} in terms of the partial quotients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , see Propositions 7 and 8. Theorems 1, 2 and 5 are proved in Sect. 2.2. In Sect. 3, we show how Theorems 3 and 4 follow from classical results on the metric theory of continued fractions and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} -mixing random variables. The proof of Theorem 6 in Sect. 4, on the other hand, relies on recent results of Bettin and Drappeau [4] on the statistics of partial quotients of random rationals.

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			\begin{document}$$L^2$$\end{document}L2 discrepancy via the Parseval formula

The main estimates

We remind that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = [a_0;a_1,a_2, \dots ]$$\end{document} is the (finite or infinite) continued fraction expansion of a real number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_k/q_k=[a_0;a_1,\dots , a_k]$$\end{document} denotes its convergents. For the rest of the paper, we also use the notation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T_n=\sum _{\ell =0}^n \left( \frac{1}{2} - \{ \ell \alpha \} \right) \quad \text {and} \quad E_N=\frac{1}{N}\sum _{n=0}^{N-1} T_n. \end{aligned}$$\end{document}

Finally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta $$\end{document} is the Riemann zeta function.

Our main tool is an evaluation of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} discrepancy up to a small error, based on the Parseval formula. This method goes back to Davenport [12], and more recently has also been used in [5–7, 18, 25]. We follow the steps in our previous paper [9], where we considered irrationals whose sequence of partial quotients is reasonably well-behaved (e.g. bounded, or increasing at a regular rate such as for Euler’s number). Here we shall need a more refined analysis in order to study arbitrary reals without any assumption on the partial quotients.

Proposition 7

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} \left| D_2^2 (S(\alpha , N)) - \sum _{m=1}^{q_{K-1}-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} - \xi _S(\alpha , N) \right| \\ {}{} & {} \quad \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2q_k} + \frac{\zeta (3)}{16 \pi ^4 N} \sum _{k=0}^{K-2} (a_{k+1}+2)^3 q_k + 6.28 \end{aligned}$$\end{document}

with some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _S(\alpha , N)$$\end{document} which satisfies both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \xi _S(\alpha , N) \le \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{2 \pi ^4\,m^2 \Vert m \alpha \Vert ^2}$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \xi _S(\alpha , N) - \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \right| \le \frac{\zeta (3)}{16 \pi ^4 N} (a_K+2)^3 q_{K-1} +0.07. \end{aligned}$$\end{document}

Similarly, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| D_2^2(L(\alpha , N)) - \frac{1}{N} \sum _{n=0}^{N-1} \left( T_n^2 + \frac{1}{2}T_n \right) - \left( 1-\frac{1}{2N} \right) \sum _{m=1}^{q_{K-1}-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} - \xi _L(\alpha , N) \right| \\ {}&\qquad \quad \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{8 q_k} + \frac{\zeta (3)}{16 \pi ^4 N} \sum _{k=0}^{K-2} (a_{k+1}+2)^3 q_k + 2.78 \end{aligned}$$\end{document}

with some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _L(\alpha , N)$$\end{document} which satisfies both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \xi _L(\alpha , N) \le \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{2 \pi ^4\,m^2 \Vert m \alpha \Vert ^2}$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \xi _L(\alpha , N) - \left( 1-\frac{1}{2N} \right) \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \right| \le \frac{\zeta (3)}{16 \pi ^4 N} (a_K+2)^3 q_{K-1}. \end{aligned}$$\end{document}

We also prove a simpler form which is sharp up to a constant factor.

Proposition 8

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2 (S(\alpha , N)) \ll \sum _{k=1}^K a_k^2$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=q_K$$\end{document} , we also have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2 (S(\alpha , q_K)) \gg \sum _{k=1}^K a_k^2$$\end{document} , and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^K a_k^2 + \left( \sum _{k=1}^K (-1)^k a_k \right) ^2 \ll D_2^2 (L(\alpha , q_K)) \ll \sum _{k=1}^K a_k^2 + \left( \sum _{k=1}^K (-1)^k a_k \right) ^2. \end{aligned}$$\end{document}

The implied constants are universal.

We postpone the proofs to Sects. 2.3 and 2.4, and now comment on the main terms.

The contribution of the sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_n$$\end{document} can be written as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{N} \sum _{n=0}^{N-1} \left( T_n^2 + \frac{1}{2} T_n \right) = \frac{1}{N} \sum _{n=0}^{N-1} (T_n-E_N)^2 + E_N^2 + \frac{1}{2} E_N. \end{aligned}$$\end{document}

Observing a connection with Dedekind sums, Beck showed [1, p. 79 and p. 91] (see also [28]) that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} , the “expected value” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_N$$\end{document} is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} E_N = \frac{1}{12} \sum _{k=1}^K (-1)^k a_k +O \left( \max _{1 \le k \le K} a_k \right) . \end{aligned}$$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=q_K$$\end{document} , the error term can be improved to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} E_{q_K} = \frac{1}{12} \sum _{k=1}^K (-1)^k a_k +O(1) . \end{aligned}$$\end{document}

Both implied constants are universal. Generalizing results of Beck, in a recent paper [8] we proved that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \le c k^d$$\end{document} with some constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 0$$\end{document} , then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} , the “variance” is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{N} \sum _{n=0}^{N-1} (T_n-E_N)^2 = \sum _{m=1}^{q_K-1} \frac{1}{8 \pi ^4 m^2 \Vert m \alpha \Vert ^2} +O \left( \max _{|k-K| \ll \log K} a_k^2 \cdot (\log \log N)^4 \right) \end{aligned}$$\end{document}

with implied constants depending only on c and d. See also Lemma 10 below.

Finally, we will need two different evaluations of the Diophantine sum appearing in Proposition 7. On the one hand, for general \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} we have [10, p. 110], [9]

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \sum _{m=1}^{q_K-1} \frac{1}{m^2 \Vert m \alpha \Vert ^2} - \frac{\pi ^4}{90} \sum _{k=1}^K a_k^2 \right| \le 152 \sum _{k=1}^K a_k . \end{aligned}$$\end{document}

On the other hand, Beck [1, p. 176] proved that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is quadratic irrational, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \ge 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=1}^M \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} = c(\alpha ) \log M +O(1) \end{aligned}$$\end{document}

with some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(\alpha )>0$$\end{document} and an implied constant depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} .

Optimal lattices

In this section, we deduce Theorems 1, 2 and 5 from Propositions 7 and 8.

Proof of Theorem 1

Consider first the symmetrized lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} . We will show the implications

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{K} \sum _{k=1}^K a_k^2 \ll 1 \,\,{} & {} \Longrightarrow \,\, D_2(S(\alpha , N)) \ll \sqrt{\log N} \,\,\\ {}{} & {} \Longrightarrow \,\, D_2(S(\alpha , q_K)) \ll \sqrt{\log q_K} \,\, \Longrightarrow \,\, \frac{1}{K} \sum _{k=1}^K a_k^2 \ll 1. \end{aligned}$$\end{document}

Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{-1} \sum _{k=1}^K a_k^2 \ll 1$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \rightarrow \infty $$\end{document} . By Proposition 8, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2 (S(\alpha , N)) \ll \sum _{k=1}^K a_k^2 \ll K \ll \log N$$\end{document} , as claimed. The second implication is trivial. Next, assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2 (S(\alpha , q_K)) \ll \sqrt{\log q_K}$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \rightarrow \infty $$\end{document} . By Proposition 8, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^K a_k^2 \ll D_2^2 (S(\alpha , q_K)) \ll \log q_K \le \sum _{k=1}^K \log (a_k+1) \ll \sum _{k=1}^K a_k \le \sqrt{K \sum _{k=1}^K a_k^2}, \end{aligned}$$\end{document}

and the claim follows. This finishes the proof of the equivalence for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} .

Consider now the unsymmetrized lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , q_K)$$\end{document} . Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{-1} \sum _{k=1}^K a_k^2 \ll 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{-1/2} |\sum _{k=1}^K (-1)^k a_k | \ll 1$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \rightarrow \infty $$\end{document} . By Proposition 8, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (L(\alpha , q_K)) \ll \sum _{k=1}^K a_k^2 + \left( \sum _{k=1}^K (-1)^k a_k \right) ^2 \ll K \ll \log q_K, \end{aligned}$$\end{document}

as claimed. Next, assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(L(\alpha , q_K)) \ll \sqrt{\log q_K}$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \rightarrow \infty $$\end{document} . By Proposition 8, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^K a_k^2 + \left( \sum _{k=1}^K (-1)^k a_k \right) ^2\ll & {} D_2^2(L(\alpha , q_K)) \ll \log q_K \\\le & {} \sum _{k=1}^K \log (a_k+1) \ll \sum _{k=1}^K a_k \le \sqrt{K \sum _{k=1}^K a_k^2}. \end{aligned}$$\end{document}

In particular,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k=1}^K a_k^2 \ll \sqrt{K \sum _{k=1}^K a_k^2} \qquad \text {and} \qquad \left( \sum _{k=1}^K (-1)^k a_k \right) ^2 \ll \sqrt{K \sum _{k=1}^K a_k^2}. \end{aligned}$$\end{document}

The first estimate gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K a_k^2 \ll K$$\end{document} , whereas the second estimate yields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sum _{k=1}^K (-1)^k a_k)^2 \ll K$$\end{document} , as claimed. This finishes the proof of the equivalence for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , q_K)$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 5

As Proposition 8 applies to both rationals and irrationals, the proof is identical to that of Theorem 1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} be a quadratic irrational. By Proposition 7 and formula (5), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (S(\alpha , N)) = \sum _{m=1}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} +O(1)=c(\alpha ) \log N +O(1), \end{aligned}$$\end{document}

as claimed. Using also formula (3), we similarly get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (L(\alpha , N)) = \frac{3}{2} c(\alpha ) \log N + E_N^2+\frac{1}{2} E_N +O((\log \log N)^4). \end{aligned}$$\end{document}

Formula (1) shows that here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_N= \frac{A(\alpha )}{12} K+O(1)=\frac{A(\alpha )}{12 \Lambda (\alpha )} \log N +O(1)$$\end{document} , and the claim follows. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 7

Lemma 9

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = [a_0;a_1,a_2,\ldots ]$$\end{document} be the (finite or infinite) continued fraction expansion of a real number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_k/q_k=[a_0;a_1,\ldots , a_k]$$\end{document} be its convergents.

• (i)For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \ge 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=1}^{q_K-1} \frac{1}{\pi ^2 m^2 \Vert m \alpha \Vert } \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2 q_k} + 3.12. \end{aligned}$$\end{document}

• (ii)For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \ge 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=q_K}^{\infty } \frac{1}{2 \pi ^2 m^2} \min \left\{ \frac{1}{4 \Vert m \alpha \Vert ^2}, n^2 \right\} \le 1.12 \frac{n}{q_K} + 0.61 \frac{n^2}{q_K^2}. \end{aligned}$$\end{document}

• (iii)For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \ge 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge q_{K-1}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=1}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \min \left\{ \frac{1}{4 N \Vert 2 m \alpha \Vert }, 1 \right\} \le \frac{\zeta (3)}{16 \pi ^4 N} \sum _{k=0}^{K-1} (a_{k+1}+2)^3 q_k+ 0.07. \end{aligned}$$\end{document}

Proof

The proof of all three claims is based on the following simple observations. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 1$$\end{document} , or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1>1$$\end{document} . For any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \ge 1$$\end{document} let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{k,a}=[aq_k, (a+1)q_k) \cap [q_k,q_{k+1})$$\end{document} be a (possibly empty) index set. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _k=q_k \alpha -p_k$$\end{document} , and recall from the general theory of continued fractions that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(q_{k+1}+q_k) \le |\delta _k|=\Vert q_k \alpha \Vert \le 1/q_{k+1}$$\end{document} . For any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in J_{k,a}$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \alpha = mp_k/q_k + m \delta _k/q_k$$\end{document} , and here the second term is negligible as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m|\delta _k|/q_k<1/q_k$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_k$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_k$$\end{document} are relatively prime, as m runs in the index set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{k,a}$$\end{document} , the numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$mp_k$$\end{document} attain each mod \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_k$$\end{document} residue class at most once. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$mp_k \not \equiv 0, \pm 1 \pmod {q_k}$$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert m \alpha \Vert = \left\| \frac{mp_k}{q_k} + \frac{m \delta _k}{q_k} \right\| \ge \left\| \frac{mp_k}{q_k} \right\| - \frac{1}{q_k} \ge \frac{1}{2} \left\| \frac{mp_k}{q_k} \right\| . \end{aligned}$$\end{document}

Therefore for any nondecreasing function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: [2,\infty ) \rightarrow [0,\infty )$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m \in J_{k,a}} f \left( \frac{1}{\Vert m \alpha \Vert } \right)&\le 3 f \left( \frac{1}{\Vert q_k \alpha \Vert } \right) + \sum _{j=2}^{q_k-2} f \left( \frac{2}{\Vert j/q_k \Vert } \right) \nonumber \\&\le 3f \left( \frac{1}{\Vert q_k \alpha \Vert } \right) + 2 \sum _{2 \le j \le q_k/2} f \left( \frac{2q_k}{j} \right) . \end{aligned}$$\end{document}

Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 f(1/\Vert q_k \alpha \Vert )$$\end{document} is an upper bound to the contribution of the three terms for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m p_k \equiv 0, \pm 1 \pmod {q_k}$$\end{document} .

We also have the simpler estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{1 \le m < q_{k+1}} f \left( \frac{1}{\Vert m \alpha \Vert } \right) \le 2 \sum _{1 \le j \le q_{k+1}/2} f \left( \frac{1}{j \Vert q_k \alpha \Vert } \right) . \end{aligned}$$\end{document}

Indeed, consider the points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \alpha \pmod {1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le m < q_{k+1}$$\end{document} and the intervals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_j=[j \Vert q_k \alpha \Vert , (j+1)\Vert q_k \alpha \Vert )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \ge 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_j = ((j-1) \Vert q \alpha \Vert , j \Vert q_k \alpha \Vert ]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \le -1$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert (m_1-m_2) \alpha \Vert \ge \Vert q_k \alpha \Vert $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1, m_2 \in [1,q_{k+1})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1 \ne m_2$$\end{document} , each interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_j$$\end{document} contains at most one point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \alpha \pmod {1}$$\end{document} , and (7) follows.

• (i)Estimate (6) yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m \in J_{k,a}} \frac{1}{\pi ^2 m^2 \Vert m \alpha \Vert }\le & {} \frac{1}{\pi ^2 a^2 q_k^2} \left( \frac{3}{\Vert q_k \alpha \Vert } + 2 \sum _{2 \le j \le q_k/2} \frac{2q_k}{j} \right) \\\le & {} \frac{1}{\pi ^2 a^2 q_k^2} \left( 3(q_{k+1}+q_k) + 4 q_k \log \frac{q_k}{2} \right) . \end{aligned}$$\end{document}

Summing over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \ge 1$$\end{document} and2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le k \le K-1$$\end{document} leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sum _{m=1}^{q_K-1} \frac{1}{\pi ^2 m^2 \Vert m \alpha \Vert }&\le \sum _{k=0}^{K-1} \frac{3 q_{k+1} + 3q_k + 4 q_k \log (q_k/2)}{6 q_k^2} \\ {}&\le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2 q_k} + \sum _{k=0}^{K-1} \frac{3+2\log (q_k/2)}{3 q_k} \\ {}&\le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2 q_k} + \sum _{k=0}^{\infty } \frac{3+2 \log (F_{k+1}/2)}{3 F_{k+1}}, \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_k$$\end{document} is the sequence of Fibonacci numbers. The numerical value of the series in the previous line is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.1195\dots $$\end{document} , as claimed.

• (ii)Estimate (6) yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m \in J_{k,a}} \frac{1}{2 \pi ^2 m^2} \min \left\{ \frac{1}{4 \Vert m \alpha \Vert ^2}, n^2 \right\}{} & {} \le \frac{1}{2 \pi ^2 a^2 q_k^2} \left( 3n^2 + 2 \sum _{j=2}^{\infty } \min \left\{ \frac{q_k^2}{j^2},n^2 \right\} \right) \\{} & {} \le \frac{1}{2 \pi ^2 a^2 q_k^2} \left( 3n^2 + 4 n q_k \right) . \end{aligned}$$\end{document}

Note that the contribution of the terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \le j \le \lfloor q_k/n \rfloor +1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \ge \lfloor q_k/n \rfloor +2$$\end{document} is at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n q_k$$\end{document} each. Summing over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \ge 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge K$$\end{document} leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=q_K}^{\infty } \frac{1}{2 \pi ^2 m^2} \min \left\{ \frac{1}{4 \Vert m \alpha \Vert ^2}, n^2 \right\} \le \sum _{k=K}^{\infty } \frac{3n^2 + 4nq_k}{12 q_k^2}. \end{aligned}$$\end{document}

From the recursion satisfied by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_k$$\end{document} one readily sees that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K+\ell } \ge F_{\ell +1} q_K$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 0$$\end{document} , hence the right hand side of the previous formula is at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1 n/q_K + c_2n^2/q_K^2$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1=\sum _{\ell =0}^{\infty } 1/(3F_{\ell +1}) = 1.1199\dots $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_2=\sum _{\ell =0}^{\infty } 1/(4F_{\ell +1}^2)=0.6065\dots $$\end{document} , as claimed.

• (iii)The contribution of all m such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert m \alpha \Vert > 1/4$$\end{document} is negligible:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\begin{array}{c} 1 \le m \le q_K-1 \\ \Vert m \alpha \Vert > 1/4 \end{array}} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \min \left\{ \frac{1}{4 N \Vert 2 m \alpha \Vert }, 1 \right\} < \sum _{m=1}^{\infty } \frac{4}{\pi ^4 m^2} = \frac{2}{3 \pi ^2}. \end{aligned}$$\end{document}

On the other hand, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert m \alpha \Vert \le 1/4$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert 2 m \alpha \Vert = 2 \Vert m \alpha \Vert $$\end{document} , hence the contribution of all such terms is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\begin{array}{c} 1 \le m \le q_K-1 \\ \Vert m \alpha \Vert \le 1/4 \end{array}} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \min \left\{ \frac{1}{4 N \Vert 2 m \alpha \Vert }, 1 \right\} \le \sum _{m=1}^{q_K-1} \frac{1}{32 \pi ^4 N m^2 \Vert m \alpha \Vert ^3}. \end{aligned}$$\end{document}

Estimate (7) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{q_k \le m < q_{k+1}} \frac{1}{32 \pi ^4 N m^2 \Vert m \alpha \Vert ^3} \le \frac{1}{16 \pi ^4 N q_k^2} \sum _{j=1}^{\infty } \frac{1}{j^3 \Vert q_k \alpha \Vert ^3} \le \frac{\zeta (3) (a_{k+1}+2)^3 q_k}{16 \pi ^4 N}. \end{aligned}$$\end{document}

Summing over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le k \le K-1$$\end{document} , we thus obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=1}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \min \left\{ \frac{1}{4 N \Vert 2 m \alpha \Vert }, 1 \right\} \le \sum _{k=0}^{K-1} \frac{\zeta (3) (a_{k+1}+2)^3 q_k}{16 \pi ^4 N} + \frac{2}{3 \pi ^2}. \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2/(3 \pi ^2)=0.06754\dots $$\end{document} , as claimed. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 7

We give a detailed proof for the symmetrized lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} , and then indicate at the end how to modify the proof for the unsymmetrized lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} .

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(x,y)=|S(\alpha , N) \cap ([0,x) \times [0,y))|$$\end{document} denote the number of points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} which fall into the box \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,x) \times [0,y)$$\end{document} . Integrating on the strips \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1) \times [n/N,(n+1)/N)$$\end{document} separately leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (S(\alpha , N)) = \sum _{n=0}^{N-1} \int _0^1 \int _{\frac{n}{N}}^{\frac{n+1}{N}} \left( B(x,y) - 2N xy \right) ^2 \, \textrm{d}y \, \textrm{d}x = M+R+\frac{4}{9} \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} M&:=\frac{1}{N} \sum _{n=0}^{N-1} \int _0^1 \left( B \left( x, \frac{n+1}{N} \right) - 2(n+1)x \right) ^2 \, \textrm{d}x, \\ R&:=\frac{2}{N} \sum _{n=0}^{N-1} \int _0^1 \left( B \left( x, \frac{n+1}{N} \right) - 2(n+1)x \right) x \, \textrm{d}x. \end{aligned} \end{aligned}$$\end{document}

The function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B \left( x, \frac{n+1}{N} \right) -2(n+1)x = \sum _{\ell =0}^{n} \left( I_{[0,x)}(\{ \ell \alpha \}) + I_{[0,x)}(\{ -\ell \alpha \}) -2x \right) , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{[0,x)}$$\end{document} denotes the indicator function of the interval [0, x), is mean zero, and has Fourier coefficients

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \int _0^1 \left( B \left( x, \frac{n+1}{N} \right) -2(n+1)x \right) e^{-2 \pi i m x} \, \textrm{d}x&= \sum _{\ell =0}^n \frac{\cos (2 \ell m \pi \alpha )}{\pi i m} \\ {}&= \frac{1}{2 \pi i m} \left( \frac{\sin ((2n+1)m \pi \alpha )}{\sin (m \pi \alpha )} + 1 \right) . \end{aligned} \end{aligned}$$\end{document}

The Fourier coefficients of x are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^1 x e^{-2\pi i m x} \, \textrm{d}x=-1/(2 \pi i m)$$\end{document} , thus by the Parseval formula we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} R= & {} \frac{2}{N} \sum _{n=0}^{N-1} 2 \sum _{m=1}^{\infty } \frac{1}{2 \pi i m} \left( \frac{\sin ((2n+1)m \pi \alpha )}{\sin (m \pi \alpha )} + 1 \right) \cdot \frac{-1}{2 \pi i m} \\ {}= & {} \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{\infty } \frac{\sin ((2n+1) m \pi \alpha )}{\pi ^2 m^2 \sin (m \pi \alpha )} +\frac{1}{6}. \end{aligned}$$\end{document}

The Parseval formula similarly gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} M&= \frac{1}{N} \sum _{n=0}^{N-1} 2 \sum _{m=1}^{\infty } \frac{1}{4 \pi ^2 m^2} \left( \frac{(\sin ((2n+1)m \pi \alpha )}{\sin (m \pi \alpha )} + 1 \right) ^2 \\ {}&= \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{\infty } \frac{\sin ^2((2n+1)m \pi \alpha )}{2\pi ^2 m^2 \sin ^2 (m \pi \alpha )} + \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{\infty } \frac{\sin ((2n+1)m \pi \alpha )}{\pi ^2 m^2 \sin (m \pi \alpha )} + \frac{1}{12}. \end{aligned} \end{aligned}$$\end{document}

The only main term in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2 (S(\alpha , N))$$\end{document} is the first double sum in the previous formula. The double sum in R and the second double sum in the previous formula are identical. Using

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \frac{\sin ((2n+1)m \pi \alpha )}{\sin (m \pi \alpha )} \right| \le \min \left\{ \frac{1}{2 \Vert m \alpha \Vert }, 2n+1 \right\} \end{aligned}$$\end{document}

and Lemma 9 (i), they can be estimated as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \bigg | \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{\infty } \frac{2 \sin ((2n+1)m \pi \alpha )}{\pi ^2 m^2 \sin (m \pi \alpha )} \bigg |&\le \frac{1}{N} \sum _{n=0}^{N-1} \left( \sum _{m=1}^{q_K-1} \frac{1}{\pi ^2 m^2 \Vert m \alpha \Vert } + \sum _{m=q_K}^{\infty } \frac{2(2n+1)}{\pi ^2 m^2} \right) \\ {}&\le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2q_k} +3.12+ \frac{4N}{\pi ^2 q_K}. \end{aligned} \end{aligned}$$\end{document}

By the assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \le q_K$$\end{document} and the fact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.12+4/\pi ^2+4/9+1/6+1/12 <4.22$$\end{document} , we thus obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| D_2^2 (S(\alpha , N)) - \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{\infty } \frac{\sin ^2((2n+1)m \pi \alpha )}{2\pi ^2 m^2 \sin ^2 (m \pi \alpha )} \right| \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2q_k} + 4.22. \end{aligned}$$\end{document}

Lemma 9 (ii) estimates the tail of the infinite series in the previous formula as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=q_K}^{\infty } \frac{\sin ^2((2n+1)m \pi \alpha )}{2\pi ^2 m^2 \sin ^2 (m \pi \alpha )}{} & {} \le \sum _{m=q_K}^{\infty } \frac{1}{2 \pi ^2 m^2} \min \left\{ \frac{1}{4 \Vert m \alpha \Vert ^2}, (2n+1)^2 \right\} \\ {}{} & {} \le 1.12 \frac{2n+1}{q_K} + 0.61 \frac{(2n+1)^2}{q_K^2}. \end{aligned}$$\end{document}

By the assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \le q_K$$\end{document} and the facts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=0}^{N-1}(2n+1)^2 \le (4/3)N^3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4.22+1.12+(4/3)\cdot 0.61<6.16$$\end{document} , we immediately get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| D_2^2 (S(\alpha , N)) - \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{q_K-1} \frac{\sin ^2((2n+1)m \pi \alpha )}{2\pi ^2 m^2 \sin ^2 (m \pi \alpha )} \right| \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2q_k} + 6.16. \end{aligned}$$\end{document}

Elementary calculations show that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\sin ^2 (\pi x) - 1/(\pi ^2 \Vert x \Vert ^2)$$\end{document} is increasing on (0, 1/2], hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(\pi ^2 \Vert x \Vert ^2) \le 1/\sin ^2 (\pi x) \le 1/(\pi ^2 \Vert x \Vert ^2) + 1-4/\pi ^2$$\end{document} for all x. The error of replacing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sin ^2 (m \pi \alpha )$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^2 \Vert m \alpha \Vert ^2$$\end{document} in the denominator of the previous formula is thus at most

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{q_K-1} \frac{\sin ^2 ((2n+1)m \pi \alpha ) (1-4/\pi ^2)}{2 \pi ^2 m^2} \le \sum _{m=1}^{\infty } \frac{1-4/\pi ^2}{2 \pi ^2 m^2} = \frac{1-4/\pi ^2}{12}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$6.16+(1-4/\pi ^2)/12 <6.21$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| D_2^2 (S(\alpha , N)) - \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{q_{K-1}-1} \frac{\sin ^2((2n+1)m \pi \alpha )}{2\pi ^4 m^2 \Vert m \alpha \Vert ^2} - \xi _S(\alpha , N) \right| \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2q_k} + 6.21, \end{aligned}$$\end{document}

where we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \xi _S(\alpha , N) :=\frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=q_{K-1}}^{q_K-1} \frac{\sin ^2((2n+1)m \pi \alpha )}{2\pi ^4 m^2 \Vert m \alpha \Vert ^2}. \end{aligned}$$\end{document}

Using the trigonometric identity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{N} \sum _{n=0}^{N-1} \sin ^2 ((2n+1)x) = \frac{1}{2} - \frac{\sin (4Nx)}{4N\sin (2x)}, \end{aligned}$$\end{document}

the double sum in (8) simplifies to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{m=1}^{q_{K-1}-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} - \sum _{m=1}^{q_{K-1}-1} \frac{\sin (4Nm \pi \alpha )}{8 \pi ^4 N m^2 \Vert m \alpha \Vert ^2 \sin (2m\pi \alpha )}. \end{aligned}$$\end{document}

Here second term can be estimated using Lemma 9 (iii) as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \left| \sum _{m=1}^{q_{K-1}-1} \frac{\sin (4Nm \pi \alpha )}{8 \pi ^4 N m^2 \Vert m \alpha \Vert ^2 \sin (2m\pi \alpha )} \right|&\le \sum _{m=1}^{q_{K-1}-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \min \left\{ \frac{1}{4N \Vert 2m \alpha \Vert }, 1 \right\} \\ {}&\le \frac{\zeta (3)}{16 \pi ^4 N} \sum _{k=0}^{K-2} (a_{k+1}+2)^3 q_k + 0.07. \end{aligned} \end{aligned}$$\end{document}

Therefore (8) simplifies to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| D_2^2 (S(\alpha , N)) - \sum _{m=1}^{q_{K-1}-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} - \xi _S(\alpha , N) \right|{} & {} \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{2q_k} \\ {}{} & {} \quad + \frac{\zeta (3)}{16 \pi ^4 N} \sum _{k=0}^{K-2} (a_{k+1}+2)^3 q_k + 6.28, \end{aligned}$$\end{document}

and it remains to prove the properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _S(\alpha , N)$$\end{document} . Clearly, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le \xi _S(\alpha , N) \le \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{2 \pi ^4\,m^2 \Vert m \alpha \Vert ^2}$$\end{document} . On the other hand, repeating arguments from above and from Lemma 9 (iii), we can also write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \xi _S(\alpha , N) = \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} - \sum _{m=q_{K-1}}^{q_K-1} \frac{\sin (4Nm \pi \alpha )}{8 \pi ^4 N m^2 \Vert m \alpha \Vert ^2 \sin (2m\pi \alpha )}, \end{aligned}$$\end{document}

consequently

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \left| \xi _S(\alpha , N) - \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \right|&\le \sum _{m=q_{K-1}}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} \min \left\{ \frac{1}{4N \Vert 2m \alpha \Vert }, 1 \right\} \\ {}&\le \frac{\zeta (3)}{16 \pi ^4 N} (a_K+2)^3 q_{K-1} +0.07. \end{aligned} \end{aligned}$$\end{document}

This finishes the proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha , N)$$\end{document} .

The proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\alpha , N)$$\end{document} is entirely analogous. The only difference is that the number of points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(x,y):=|L(\alpha , N) \cap ([0,x) \times [0,y))|$$\end{document} which fall into the box \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,x) \times [0,y)$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B \left( x, \frac{n+1}{N} \right) - (n+1) x = \sum _{\ell =0}^n \left( I_{[0,x)}(\{ \ell \alpha \}) -x \right) , \end{aligned}$$\end{document}

which is not a mean zero function. Its integral (0th Fourier coefficient) is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^1 \left( B \left( x, \frac{n+1}{N} \right) - (n+1) x \right) \, \textrm{d} x = \sum _{\ell =0}^n \left( \frac{1}{2} - \{ \ell \alpha \} \right) =T_n, \end{aligned}$$\end{document}

which introduces the extra terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{-1} \sum _{n=0}^{N-1} T_n/2$$\end{document} resp. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{-1} \sum _{n=0}^{N-1} T_n^2$$\end{document} when the Parseval formula is applied to the analogue of R resp. M as above. For the convenience of the reader we mention that the analogue of formula (8) is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| D_2^2 (L(\alpha , N)) - \frac{1}{N} \sum _{n=0}^{N-1} \left( T_n^2+\frac{1}{2} T_n \right) - \frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=1}^{q_{K-1}-1} \frac{\sin ^2((n+1)m \pi \alpha )}{2\pi ^4 m^2 \Vert m \alpha \Vert ^2} - \xi _L(\alpha , N) \right| \\ \le \sum _{k=0}^{K-1} \frac{a_{k+1}}{8q_k} + 2.78, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \xi _L (\alpha , N) :=\frac{1}{N} \sum _{n=0}^{N-1} \sum _{m=q_{K-1}}^{q_K-1} \frac{\sin ^2((n+1)m \pi \alpha )}{2\pi ^4 m^2 \Vert m \alpha \Vert ^2}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 8

The following lemma is a simpler form of formula (3), but it applies without any assumption on the partial quotients. As modifying the proof of (3) is not entirely straightforward, we include the details.

Lemma 10

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \ge 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{q_K} \sum _{n=0}^{q_K-1} (T_n-E_{q_K})^2 \ll \sum _{k=1}^K a_k^2 \end{aligned}$$\end{document}

with a universal implied constant.

Proof

For the sake of readability, set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=p_K$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=q_K$$\end{document} . For any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le \ell \le q-1$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \ell p/q \Vert \ge 1/q$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell \alpha - \ell p/q| \le q |\alpha -p/q| < 1/q$$\end{document} . Thus there is no integer between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell p/q$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \alpha $$\end{document} , hence

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \{ \ell \alpha \} - \left\{ \frac{\ell p}{q} \right\} \right| \le \left| \ell \alpha - \frac{\ell p}{q} \right| < \frac{1}{q}. \end{aligned}$$\end{document}

Consequently, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le n \le q-1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T_n=\sum _{\ell =0}^n \left( \frac{1}{2} - \{ \ell \alpha \} \right) = \sum _{\ell =0}^n \left( \frac{1}{2} - \frac{1}{2q} - \left\{ \frac{\ell p}{q} \right\} \right) + O(1). \end{aligned}$$\end{document}

Introducing

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T_n^* :=\sum _{\ell =0}^n \left( \frac{1}{2} - \frac{1}{2q} - \left\{ \frac{\ell p}{q} \right\} \right) \quad \text {and} \quad E_q^* :=\frac{1}{q} \sum _{n=0}^{q-1} T_n^*, \end{aligned}$$\end{document}

we thus have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_n -E_q = T_n^* - E_q^*+O(1)$$\end{document} . Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{-1} \sum _{n=0}^{q-1} (T_n-E_q)^2 \ll q^{-1} \sum _{n=0}^{q-1} (T_n^*-E_q^*)^2 +1$$\end{document} , and it remains to estimate the latter.

The rest of the proof is based on Fourier analysis on the finite cyclic group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_q$$\end{document} , which we identify by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ 0,1,\dots , q-1 \}$$\end{document} . Elementary calculations show that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{x=0}^{q-1} \left( \frac{1}{2} - \frac{1}{2q} - \left\{ \frac{x}{q} \right\} \right) e^{-2 \pi i m x/q} = \left\{ \begin{array}{ll} 0 &{} \text {if } m=0, \\ 1/(1-e^{-2 \pi i m/q}) &{} \text {if } 1 \le m \le q-1. \end{array} \right. \end{aligned}$$\end{document}

Therefore by Fourier inversion on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_q$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{2} - \frac{1}{2q} - \left\{ \frac{x}{q} \right\} = \frac{1}{q} \sum _{m=1}^{q-1} \frac{e^{2 \pi i m x/q}}{1-e^{-2 \pi i m /q}}, \qquad x \in \mathbb {Z}. \end{aligned}$$\end{document}

We can thus write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_n^*$$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T_n^* = \frac{1}{q} \sum _{m=1}^{q-1} \sum _{\ell =0}^n \frac{e^{2 \pi i m \ell p /q}}{1-e^{-2 \pi i m/q}} = \frac{1}{q} \sum _{m=1}^{q-1} \frac{1-e^{2 \pi i m (n+1)p/q}}{(1-e^{- 2 \pi i m/q}) (1-e^{2 \pi i m p/q})}. \end{aligned}$$\end{document}

Letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B=q^{-1} \sum _{m=1}^{q-1} 1/(1-e^{-2 \pi i m/q})(1-e^{2 \pi i mp/q})$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{q} \sum _{n=0}^{q-1} (T_n^*-E_q^*)^2 \le \frac{1}{q} \sum _{n=0}^{q-1} |T_n^*-B|^2 = \frac{1}{q} \sum _{n=0}^{q-1} \frac{1}{q^2} \left| \sum _{m=1}^{q-1} \frac{e^{2 \pi i m (n+1)p/q}}{(1-e^{-2 \pi i m/q})(1-e^{2 \pi i m p/q})} \right| ^2. \end{aligned}$$\end{document}

Expanding the square shows that here

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\bigg | \sum _{m=1}^{q-1} \frac{e^{2 \pi i m (n+1)p/q}}{(1-e^{-2 \pi i m/q})(1-e^{2 \pi i m p/q})} \bigg |^2 \\ {}&\quad = \sum _{m=1}^{q-1} \frac{1}{|1-e^{-2 \pi i m/q}|^2 |1-e^{2 \pi i m p/q}|^2} \\ {}&\qquad + \sum _{\begin{array}{c} m_1, m_2=1 \\ m_1 \ne m_2 \end{array}}^{q-1} \frac{e^{2 \pi i (m_1-m_2)(n+1)p/q}}{(1-e^{- 2 \pi i m_1 /q})(1-e^{2 \pi i m_1 p/q})(1-e^{2 \pi i m_2 /q})(1-e^{-2 \pi i m_2 p/q})}. \end{aligned} \end{aligned}$$\end{document}

As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=0}^{q-1} e^{2 \pi i (m_1-m_2)(n+1)p/q} =0$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_1 \ne m_2$$\end{document} , the contribution of the off-diagonal terms is zero. Formula (4) thus leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{q} \sum _{n=0}^{q-1} (T_n^*-E_q^*)^2 \le \frac{1}{q^2} \sum _{m=1}^{q-1} \frac{1}{|1-e^{-2 \pi i m/q}|^2 |1-e^{2 \pi i mp/q}|^2} \ll \sum _{m=1}^{q-1} \frac{1}{m^2 \Vert mp/q \Vert ^2} \ll \sum _{k=1}^K a_k^2, \end{aligned}$$\end{document}

as claimed. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 8

By Proposition 7, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2(S(\alpha , N)) \ll \sum _{m=1}^{q_K-1} \frac{1}{m^2 \Vert m \alpha \Vert ^2} + \sum _{k=0}^{K-1} \frac{a_{k+1}}{q_k} + \sum _{k=0}^{K-2} \frac{a_{k+1}^3 q_k}{N}. \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k+1}^3 q_k /N \le a_{k+1}^2$$\end{document} , hence formula (4) yields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(S(\alpha , N)) \ll \sum _{k=1}^K a_k^2$$\end{document} , as claimed. Using Lemma 10 and formula (2) we also deduce that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=q_K$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{q_K} \sum _{n=0}^{q_K-1} \left( T_n^2+\frac{1}{2} T_n \right) = \frac{1}{q_K} \sum _{n=0}^{q_K-1} (T_n-E_{q_K})^2 + E_{q_K}^2 + \frac{1}{2} E_{q_K} \ll \sum _{k=1}^K a_k^2 + \left( \sum _{k=1}^K (-1)^k a_k \right) ^2, \end{aligned}$$\end{document}

and the upper bound for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(L(\alpha , q_K))$$\end{document} follows.

Next, we prove the lower bounds. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>0$$\end{document} resp. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} denote suitably small resp. large universal constants whose values change from line to line. By Proposition 7 and formula (4), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=q_K$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} D_2^2 (S(\alpha , q_K))&\ge \sum _{m=1}^{q_K-1} \frac{1}{4 \pi ^4 m^2 \Vert m \alpha \Vert ^2} - \frac{\zeta (3)}{16 \pi ^4 q_K} \sum _{k=0}^{K-1} (a_{k+1}+2)^3 q_k - C \sum _{k=1}^K a_k \\ {}&\ge \left( \frac{1}{360} - \frac{\zeta (3)}{16 \pi ^4} \right) \sum _{k=1}^K a_k^2 -C \sum _{k=1}^K a_k. \end{aligned} \end{aligned}$$\end{document}

The point is that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/360>\zeta (3)/(16 \pi ^4)$$\end{document} , i.e. the coefficient of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k^2$$\end{document} is positive. The contribution of all k such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \ll 1$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll K$$\end{document} , and for all other terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k^2$$\end{document} dominates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k$$\end{document} . Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2 (S(\alpha , q_K)) \ge c \sum _{k=1}^K a_k^2 -CK$$\end{document} . On the other hand, by Roth’s theorem we also have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2 (S(\alpha , q_K)) \gg \log q_K \gg K$$\end{document} . Taking a suitable weighted average of the previous two inequalities establishes the lower bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(S(\alpha , q_K)) \ge c \sum _{k=1}^K a_k^2$$\end{document} .

From Proposition 7 we similarly deduce

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2(L(\alpha , q_K)) \ge \frac{1}{q_K} \sum _{n=1}^{q_K-1} \left( T_n-E_{q_K} \right) ^2 + E_{q_K}^2 + c \sum _{k=1}^K a_k^2. \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_K^{-1} \sum _{n=0}^{q_K-1} (T_n-E_{q_K})^2 \ge 0$$\end{document} , and the lower bound for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(L(\alpha , q_K))$$\end{document} follows from formula (2). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Remark 1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} be an irrational such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \ll \sqrt{k}/(\log k)^2$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} we then have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{|k-K| \ll \log K} a_k^2 \cdot (\log \log N)^4 \ll K$$\end{document} , hence formulas (3) and (4) give

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{N} \sum _{n=0}^{N-1} (T_n-E_N)^2 = \sum _{m=1}^{q_K-1} \frac{1}{8 \pi ^4 m^2 \Vert m \alpha \Vert ^2} +O(K) \ll \sum _{k=1}^K a_k^2. \end{aligned}$$\end{document}

Using this fact instead of Lemma 10 in the proof of Proposition 8, we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(L(\alpha , N)) \ll \sum _{k=1}^K a_k^2 + (\sum _{k=1}^K (-1)^k a_k)^2$$\end{document} holds for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K-1} \le N \le q_K$$\end{document} (instead of only for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=q_K$$\end{document} ). In particular, the equivalence stated in Remark 1 follows.

Typical irrationals

Asymptotics almost everywhere

Let us recall certain basic facts about the statistics of the partial quotients of a typical irrational number. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} be a positve nondecreasing function on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\infty )$$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_K=\max _{1 \le k \le K} a_k$$\end{document} . It is well known that for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log q_k \sim \frac{\pi ^2}{12 \log 2} k$$\end{document} , and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k \le \varphi (k)$$\end{document} for all but finitely many k if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty } 1/\varphi (n)< \infty $$\end{document} . A classical result of Diamond and Vaaler [13] on trimmed sums states that for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\sum _{k=1}^K a_k - A_K}{K \log K} \rightarrow \frac{1}{\log 2} \qquad \text {as } K \rightarrow \infty . \end{aligned}$$\end{document}

Proof of Theorem 3

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 2$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N(\alpha )$$\end{document} be the positive integer for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K_N(\alpha ) -1} < N \le q_{K_N(\alpha )}$$\end{document} . In particular, for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N(\alpha ) \sim \frac{12 \log 2}{\pi ^2} \log N$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{12 \log 2}{\pi ^2}=0.8427\dots $$\end{document} .

(i) Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty } 1/\varphi (n)<\infty $$\end{document} . As observed in the Introduction, by a classical discrepancy estimate for the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ n \alpha \}$$\end{document} [16, p. 52], we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&D_2 (S(\alpha , N)) \ll D_{\infty } (L(\alpha , N)) \ll \sum _{k=1}^{K_N(\alpha )} a_k, \\ {}&D_2 (L(\alpha , N)) \ll D_{\infty } (L(\alpha , N)) \ll \sum _{k=1}^{K_N(\alpha )} a_k. \end{aligned} \end{aligned}$$\end{document}

The asymptotic relation (9) of Diamond and Vaaler shows that for a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&D_2 (S(\alpha , N)) \le C \sum _{k=1}^{K_N(\alpha )} a_k = C A_{K_N(\alpha )} + O(K_N(\alpha ) \log K_N(\alpha )), \\ {}&D_2 (L(\alpha , N)) \le C \sum _{k=1}^{K_N(\alpha )} a_k = C A_{K_N(\alpha )} + O(K_N(\alpha ) \log K_N(\alpha ) ) \end{aligned} \end{aligned}$$\end{document}

with a universal constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} . Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{K_N(\alpha )} \le \varphi (K_N(\alpha ))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N(\alpha ) \le \log N$$\end{document} for all but finitely many N. Therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(S(\alpha , N)) \le C \varphi (\log N)+O(\log N \log \log N)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(L(\alpha , N)) \le C \varphi (\log N)+O(\log N \log \log N)$$\end{document} with implied constants depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} . The factor C can be removed by repeating the argument with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x)/C$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x)$$\end{document} .

(ii) Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty } 1/\varphi (n)=\infty $$\end{document} . By Proposition 8, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2(S(\alpha , q_K)) \ge c \bigg ( \sum _{k=1}^K a_k^2 \bigg )^{1/2} \ge c A_K \quad \text {and} \quad D_2(L(\alpha , q_K)) \ge c \bigg ( \sum _{k=1}^K a_k^2 \bigg )^{1/2} \ge c A_K \end{aligned}$$\end{document}

with a universal constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>0$$\end{document} . Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_K \ge \varphi (K)$$\end{document} for infinitely many K, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \ge (\log q_K)/2$$\end{document} for all but finitely many K. Hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(S(\alpha , q_K)) \ge c \varphi ((\log q_K)/2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(L(\alpha , q_K)) \ge c \varphi ((\log q_K)/2)$$\end{document} for infinitely many K. Repeating the argument with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (2x)/c$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x)$$\end{document} , we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(S(\alpha , q_K)) \ge \varphi (\log q_K)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2(L(\alpha , q_K)) \ge \varphi (\log q_K)$$\end{document} for infinitely many K, as claimed. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Limit distribution

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} be the Lebesgue measure, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu (B)= (1/\log 2) \int _B 1/(1+x) \, \textrm{d} x$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B \subseteq [0,1]$$\end{document} Borel) the Gauss measure. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is chosen randomly from [0, 1] with distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , then its partial quotients are identically distributed random variables with distribution

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \,: \, a_k = n \right\} \right) = \frac{1}{\log 2} \log \left( 1+\frac{1}{n(n+2)} \right) , \qquad k,n \ge 1. \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is chosen randomly from [0, 1] with distribution either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , then the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} -mixing with exponential rate [21, p. 119].

To find the limit distribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(S(\alpha , N))/(\log N)^2$$\end{document} , we shall need more sophisticated facts about the partial quotients of a typical irrational, which we now gather. Most importantly, a special case of a limit distribution theorem of Samur [27] (see also [8]) states that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is a Borel probability measure on [0, 1] which is absolutely continuous with respect to the Lebesgue measure, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu \left( \left\{ \alpha \in [0,1] \, : \, \frac{2 (\log 2)^2}{\pi K^2} \sum _{k=1}^K a_k^2 \le t \right\} \right) \rightarrow \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d}x \qquad \text {as } K \rightarrow \infty . \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , then general results of Heinrich [17] on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} -mixing random variables imply the rate of convergence

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \ge 0} \left| \mu \left( \left\{ \alpha \in [0,1] \, : \, \frac{2 (\log 2)^2}{\pi K^2} \sum _{k=1}^K a_k^2 \le t \right\} \right) - \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d}x \right| \ll \frac{1}{K^{1-\varepsilon }} \end{aligned}$$\end{document}

with an arbitrary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} and an implied constant depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . The corresponding result for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K a_k$$\end{document} in the Gauss measure is also due to Heinrich:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \in \mathbb {R}} \left| \nu \left( \left\{ \alpha \in [0,1] \,: \, \frac{1}{K} \sum _{k=1}^K a_k - \frac{\log K -\gamma }{\log 2} \le t \right\} \right) - F(t) \right| \ll \frac{(\log K)^2}{K}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is the Euler–Mascheroni constant, and F(t) is the distribution function of the law with characteristic function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\mathbb {R}} e^{itx} \, \textrm{d}F(t) = \exp \left( - \frac{\pi }{2 \log 2} |x| \left( 1+\frac{2i}{\pi } \textrm{sgn}(x) \log |x| \right) \right) . \end{aligned}$$\end{document}

Note that this is a stable law with stability parameter 1 (and skewness parameter 1). Hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-F(t) \ll 1/t$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow \infty $$\end{document} , and we immediately obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \, : \, \frac{1}{K} \sum _{k=1}^K a_k \ge t + \frac{\log K}{\log 2} \right\} \right) \ll \frac{1}{t} + \frac{(\log K)^2}{K} \qquad \text {as } t \rightarrow \infty . \end{aligned}$$\end{document}

The final ingredient is a similar estimate for the convergent denominators: with a large enough universal constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \, : \, \left| \log q_K - \frac{\pi ^2}{12 \log 2} K \right| \ge C \sqrt{K \log K} \right\} \right) \ll \frac{1}{\sqrt{K}} . \end{aligned}$$\end{document}

This follows from the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log q_K$$\end{document} satisfies the central limit theorem with rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/\sqrt{K})$$\end{document} , as shown by Morita [23]. We mention that a better upper bound can be deduced from the large deviation inequality of Takahasi [29], but (13) suffices for our purposes.

Proof of Theorem 4

Throughout the proof, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} is a large universal constant whose value changes from line to line, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i=Y_i(\alpha , N)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2,\ldots $$\end{document} are error terms. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 2$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N(\alpha )$$\end{document} be the positive integer for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{K_N(\alpha ) -1} < N \le q_{K_N(\alpha )}$$\end{document} .

Proposition 7 and formula (4) show that we can write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (S(\alpha , N))&= \dfrac{1}{360} \sum \limits _{k=1}^{K_N(\alpha )-1} a_k^2 + Y_1, \quad \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |Y_1| \le \frac{1}{180} a_{K_N(\alpha )}^2+ C \sum _{k=1}^{K_N(\alpha )} a_k + \frac{C}{N} \sum _{k=0}^{K_N(\alpha ) -2} a_{k+1}^3 q_k. \end{aligned}$$\end{document}

Using the general fact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k+2}/q_k \ge 2$$\end{document} , we estimate the last error term as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \frac{1}{N} \sum _{k=0}^{K_N(\alpha )-2} a_{k+1}^3 q_k&\le \frac{1}{N} \sum _{k=1}^{K_N(\alpha )-1} a_k^2 q_k \\ {}&\le \sum _{k=1}^{K_N(\alpha ) - 100 \log K_N(\alpha )} a_k^2 \frac{q_k}{q_{K_N(\alpha )-1}} + \sum _{k=K_N(\alpha ) - 100 \log K_N(\alpha )}^{K_N(\alpha ) -1} a_k^2 \frac{q_k}{q_{K_N(\alpha ) -1}} \\ {}&\le \frac{1}{K_N(\alpha )^{10}} \sum _{k=1}^{K_N(\alpha )} a_k^2 + \sum _{k=K_N(\alpha ) - 100 \log K_N(\alpha )}^{K_N(\alpha ) -1} a_k^2. \end{aligned} \end{aligned}$$\end{document}

This leads to the simplified form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(S(\alpha , N)) = (1/360) \sum _{k=1}^{K_N(\alpha )} a_k^2 + Y_2$$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |Y_2| \le \frac{C}{K_N(\alpha )^{10}} \sum _{k=1}^{K_N(\alpha )} a_k^2 + C\sum _{k=K_N(\alpha ) -100 \log K_N(\alpha )}^{K_N(\alpha )} a_k^2 + C\sum _{k=1}^{K_N(\alpha )} a_k. \end{aligned}$$\end{document}

Set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K}=\lceil \frac{12 \log 2}{\pi ^2} \log N \rceil $$\end{document} . The estimate (13) states that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \,: \, \left| \log q_{\overline{K}} - \frac{\pi ^2}{12 \log 2} \overline{K} \right| \ge C \sqrt{\overline{K} \log \overline{K}} \right\} \right) \ll \frac{1}{\sqrt{\overline{K}}}. \end{aligned}$$\end{document}

By the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N(\alpha )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K}$$\end{document} , this immediately gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \,: \, |K_N(\alpha ) - \overline{K}| \ge C \sqrt{\overline{K} \log \overline{K}} \right\} \right) \ll \frac{1}{\sqrt{\overline{K}}}. \end{aligned}$$\end{document}

Roughly speaking, this means that we can replace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N(\alpha )$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K}$$\end{document} in the above formulas; the point is that the latter does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} . More precisely, outside a set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} -measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll 1/\sqrt{\overline{K}}$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2^2(S(\alpha , N)) = (1/360) \sum _{k=1}^{\overline{K}} a_k^2+Y_3$$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |Y_3| \le \frac{C}{\overline{K}^{10}} \sum _{k=1}^{2 \overline{K}} a_k^2 + C \sum _{k=\overline{K}-C\sqrt{\overline{K} \log \overline{K}}}^{\overline{K} + C \sqrt{\overline{K} \log \overline{K}}} a_k^2 +C \sum _{k=1}^{2\overline{K}} a_k. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5 \pi ^3 /(\log N)^2 = 720 (\log 2)^2 /(\pi \overline{K}^2) +O(1/\overline{K}^3)$$\end{document} , normalizing the previous formula leads to the fact that outside a set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} -measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll 1/\sqrt{\overline{K}}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 5 \pi ^3 \frac{D_2^2 (S(\alpha , N))}{(\log N)^2} = \frac{2 (\log 2)^2}{\pi \overline{K}^2} \sum _{k=1}^{\overline{K}} a_k^2 +Y_4, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |Y_4| \le \frac{C}{\overline{K}^3} \sum _{k=1}^{2 \overline{K}} a_k^2 + \frac{C}{\overline{K}^2} \sum _{k=\overline{K}-C\sqrt{\overline{K} \log \overline{K}}}^{\overline{K} + C \sqrt{\overline{K} \log \overline{K}}} a_k^2 +\frac{C}{\overline{K}^2} \sum _{k=1}^{2\overline{K}} a_k. \end{aligned}$$\end{document}

We now estimate the three error terms in the previous formula. The limit distribution with rate of Heinrich (11) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \,: \, \frac{1}{\overline{K}^3} \sum _{k=1}^{2 \overline{K}} a_k^2 \ge \frac{1}{\overline{K}^{1/3}} \right\} \right) \ll \int _{\textrm{const} \cdot \overline{K}^{2/3}}^{\infty } \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x + \frac{1}{\overline{K}^{1-\varepsilon }} \ll \frac{1}{\overline{K}^{1/3}}. \end{aligned}$$\end{document}

Since the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k$$\end{document} is strictly stationary, we similarly deduce

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\nu \Bigg ( \Bigg \{ \alpha \in [0,1] \,: \, \frac{1}{\overline{K}^2} \sum _{k=\overline{K}-C \sqrt{\overline{K} \log \overline{K}}}^{\overline{K}+C \sqrt{\overline{K} \log \overline{K}}} a_k^2 \ge \frac{(\log \overline{K})^{1/3}}{\overline{K}^{1/3}} \Bigg \} \Bigg ) \\ {}&\quad = \nu \left( \left\{ \alpha \in [0,1] \,: \, \frac{1}{\overline{K}^2} \sum _{k=1}^{C \sqrt{\overline{K} \log \overline{K}}} a_k^2 \ge \frac{(\log \overline{K})^{1/3}}{\overline{K}^{1/3}} \right\} \right) \\ {}&\quad \ll \int _{\textrm{const} \cdot \overline{K}^{2/3}/(\log \overline{K})^{2/3}}^{\infty } \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x + \frac{1}{\overline{K}^{1/2-\varepsilon }} \\ {}&\quad \ll \frac{(\log \overline{K})^{1/3}}{\overline{K}^{1/3}}. \end{aligned} \end{aligned}$$\end{document}

Finally, formula (12) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \,: \, \frac{1}{\overline{K}^2} \sum _{k=1}^{2\overline{K}} a_k \ge \frac{1}{\overline{K}^{1/3}} \right\} \right) \ll \frac{1}{\overline{K}^{2/3}}. \end{aligned}$$\end{document}

By the previous three estimates, we can finally write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 5 \pi ^3 \frac{D_2^2 (S(\alpha , N))}{(\log N)^2} = \frac{2 (\log 2)^2}{\pi \overline{K}^2} \sum _{k=1}^{\overline{K}} a_k^2 +Y_5, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu \left( \left\{ \alpha \in [0,1] \, : \, |Y_5| \ge C \frac{(\log \overline{K})^{1/3}}{\overline{K}^{1/3}} \right\} \right) \le C \frac{(\log \overline{K})^{1/3}}{\overline{K}^{1/3}} . \end{aligned}$$\end{document}

The proof of the theorem is now immediate. Assume first, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is absolutely continuous with respect to the Lebesgue measure. The theorem of Samur (10) ensures that the main term in (14) converges in distribution to the standard Lévy distribution as N, and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{K}$$\end{document} , goes to infinity. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_5 \rightarrow 0$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} -measure, the same holds also in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} -measure, and the convergence to the standard Lévy distribution remains true for the left hand side of (14). This finishes the proof for a general absolutely continuous measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} .

Next, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} be either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} . Then the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} -mixing with exponential rate, and the limit distribution with rate of Heinrich (11) ensures that the main term in (14) converges to the standard Lévy distribution with rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll 1/\overline{K}^{1-\varepsilon }$$\end{document} . The estimate (15), which holds also with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} in place of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , together with the trivial fact that the distribution function of the Lévy distribution is Lipschitz, shows that this convergence remains true for the left hand side of (14) with the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll (\log \overline{K})^{1/3} / \overline{K}^{1/3}$$\end{document} . This finishes the proof of the rate of convergence for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Typical rationals

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_Q$$\end{document} denote the set of all reduced fractions in the interval (0, 1) with denominator at most Q, and let us write every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/q \in \mathcal {F}_Q$$\end{document} in the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/q=[0;a_1, \ldots , a_r]$$\end{document} . It does not matter which of the two possible expansions is chosen. Note that the partial quotients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1=a_1(p/q), \ldots , a_r=a_r(p/q)$$\end{document} as well as the length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=r(p/q)$$\end{document} are functions of p/q. For the sake of simplicity, we use the convention \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k=0$$\end{document} if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>r$$\end{document} .

The proof of Theorem 6 is based on recent results of Bettin and Drappeau on the limit distribution of power sums of the partial quotients; they are perfect analogues of the results for typical irrationals mentioned in Sect. 3.2.

Lemma 11

(Bettin–Drappeau [4]) For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \ge 2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \ge 0} \left| \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \, : \, \frac{\pi ^3}{72 (\log Q)^2} \sum _{k=1}^r a_k^2 \le t \right\} \right| - \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x \right| \ll \frac{1}{(\log Q)^{1-\varepsilon }} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \in \mathbb {R}} \left| \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, \frac{1}{\log Q} \sum _{k=1}^r a_k - \frac{\log \log Q - \gamma }{\pi ^2/12} \le t \right\} \right| - G(t) \right| \ll \frac{1}{(\log Q)^{1-\varepsilon }} \end{aligned}$$\end{document}

with implied constants depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is the Euler–Mascheroni constant, and G(t) is the distribution function of the law with characteristic function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\mathbb {R}} e^{itx} \, \textrm{d} G(t) = \exp \left( - \frac{6}{\pi } |x| \left( 1 + \frac{2i}{\pi } \textrm{sgn}(x) \log |x| \right) \right) . \end{aligned}$$\end{document}

The second limit distribution in Lemma 11 immediately yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \, : \, \frac{1}{\log Q} \sum _{k=1}^r a_k \ge t + \frac{\log \log Q}{\pi ^2/12} \right\} \right| \ll \frac{1}{t} + \frac{1}{(\log Q)^{1-\varepsilon }} \quad \text {as } t \rightarrow \infty . \end{aligned}$$\end{document}

Note that (16) was stated in [4] with the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll 1/(\log \log Q)^{1-\varepsilon }$$\end{document} , but the methods of that paper actually give \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll 1/(\log Q)^{1-\varepsilon }$$\end{document} . For the sake of completeness, we deduce (16) as stated here in Sect. 4.1. We now prove a lemma which will serve as a substitute for the fact that the partial quotients are not exactly identically distributed, and then prove Theorem 6.

Lemma 12

For any positive integers Q, k, t, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, a_k \ge t \right\} \right| \le \frac{2Q^2}{t}. \end{aligned}$$\end{document}

Proof

Assume first, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} . Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1 \ge t$$\end{document} implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < p/q \le 1/t$$\end{document} . In particular, for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le q \le Q$$\end{document} there are at most q/t possible numerators p, hence

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \, : \, a_1 \ge t \right\} \right| \le \sum _{q=1}^Q \frac{q}{t} \le \frac{Q^2}{t} . \end{aligned}$$\end{document}

Next, assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 2$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{denom}(x)$$\end{document} denote the denominator of a rational x (in its reduced form). From the recursion satisfied by the denominator of the convergents one readily deduces the supermultiplicative property

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{denom}([0;a_1, \ldots , a_r]) \ge \textrm{denom}([0;a_1, \ldots ,a_{k-1}]) \cdot \textrm{denom}([0;a_k, \ldots , a_r]). \end{aligned}$$\end{document}

For any fixed positive integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1, \ldots , b_{k-1}$$\end{document} we thus obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, a_1=b_1, \ldots , a_{k-1}=b_{k-1}, \,\, a_k \ge t \right\} \right| \\ {}{} & {} \qquad \le \left| \left\{ \frac{p}{q} \in \mathcal {F}_{Q/\textrm{denom}([0;b_1, \ldots , b_{k-1}])} \,: \, a_1 \ge t \right\} \right| . \end{aligned}$$\end{document}

Summing over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1, \ldots , b_{k-1}$$\end{document} and applying (18) leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, a_k \ge t \right\} \right| \le \sum _{b_1, \ldots , b_{k-1}=1}^{\infty } \frac{Q^2}{t (\textrm{denom}([0;b_1, \dots , b_{k-1}]))^2}. \end{aligned}$$\end{document}

Recall that the set of real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0;c_1, c_2, \ldots ] \in [0,1]$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1=b_1, \ldots , c_{k-1}=b_{k-1}$$\end{document} is an interval whose length is at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(2 \, \textrm{denom}([0;b_1, \ldots , b_{k-1}])^2)$$\end{document} . Since these are pairwise disjoint intervals, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{b_1, \ldots , b_{k-1}=1}^{\infty } \frac{1}{(\textrm{denom}([0;b_1, \ldots , b_{k-1}]))^2} \le 2, \end{aligned}$$\end{document}

and the claim follows. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 6

Throughout the proof, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} is a large universal constant whose value changes from line to line, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_i=Z_i (p/q)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} are error terms.

Proposition 7 and formula (4) show that we can write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_2^2 (S(p/q,q)) = \frac{1}{360} \sum _{k=1}^r a_k^2 +Z_1, \quad \text {where} \quad |Z_1| \le C \sum _{k=1}^r a_k + \frac{C}{q} \sum _{k=0}^{r-1} a_{k+1}^3 q_k. \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{k+1}^3 q_k \le a_{k+1}^2 q_{k+1}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_k/q=q_k/q_r \le 1/F_{r-k+1}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_k$$\end{document} is the sequence of Fibonacci numbers. Hence normalizing the previous formula leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} 5 \pi ^3 \frac{D_2^2 (S(p/q,q))}{(\log Q)^2} = \frac{\pi ^3}{72 (\log Q)^2} \sum _{k=1}^r a_k^2 + Z_2, \quad \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} |Z_2| \le \frac{C}{(\log Q)^2} \sum _{k=1}^r a_k + \frac{C}{(\log Q)^2} \sum _{k=1}^r \frac{a_k^2}{F_{r-k+1}}. \end{aligned}$$\end{document}

The first error term can be estimated in measure using formula (17) as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, \frac{1}{(\log Q)^2} \sum _{k=1}^r a_k \ge \frac{1}{(\log Q)^{1/2}} \right\} \right| \ll \frac{1}{(\log Q)^{1/2}}. \end{aligned}$$\end{document}

Let us agree for the moment to use the continued fraction expansion with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_r \ge 2$$\end{document} for fractions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/q \in (0,1/2]$$\end{document} , and the expansion with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_r=1$$\end{document} for fractions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p/q \in (1/2,1)$$\end{document} . Then the map reversing the order of the partial quotients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_Q \rightarrow \mathcal {F}_Q$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0;a_1, a_2, \ldots , a_r] \mapsto [0; a_r, \ldots , a_2, a_1]$$\end{document} is a bijection. In fact, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0;a_r,\ldots , a_2,a_1]$$\end{document} is the reduced fraction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{r-1}/q_r$$\end{document} , which has the same denominator as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0;a_1,\ldots , a_r]$$\end{document} . Therefore the distribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_r, \ldots , a_2, a_1)$$\end{document} is identical to that of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_1, a_2, \ldots , a_r)$$\end{document} , and we can apply Lemma 12 to estimate the second error term in measure as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, \frac{1}{(\log Q)^2} \sum _{k=1}^r \frac{a_k^2}{F_{r-k+1}} \ge \frac{1}{(\log Q)^{1/2}} \right\} \right| \\ {}&\quad = \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, \sum _{k=1}^r \frac{a_k^2}{F_k} \ge (\log Q)^{3/2} \right\} \right| \\ {}&\quad \le \frac{1}{|\mathcal {F}_Q|} \sum _{k=1}^{\infty } \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, \frac{a_k^2}{F_k} \ge (\log Q)^{3/2} \right\} \right| \\ {}&\quad \le \frac{1}{|\mathcal {F}_Q|} \sum _{k=1}^{\infty } \frac{2Q^2}{F_k^{1/2} (\log Q)^{3/4}} \\ {}&\quad \ll \frac{1}{(\log Q)^{3/4}}. \end{aligned} \end{aligned}$$\end{document}

Note that we used the convention \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_k=0$$\end{document} if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>r$$\end{document} , and the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathcal {F}_Q| \gg Q^2$$\end{document} . One readily checks that the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^r a_k^2 / F_{r-k+1}$$\end{document} for the two possible continued fraction expansions of the same rational differ at most by a factor of 2. Hence the tail estimate in the previous formula holds no matter which expansion we choose. In particular,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, |Z_2| \ge \frac{1}{(\log Q)^{1/2}} \right\} \right| \ll \frac{1}{(\log Q)^{1/2}}, \end{aligned}$$\end{document}

and the limit distribution theorem (16) of Bettin and Drappeau yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \ge 0} \left| \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, 5 \pi ^3 \frac{D_2^2 (S(p/q,q))}{(\log Q)^2} \le t \right\} \right| - \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x \right| \ll \frac{1}{(\log Q)^{1/2}}. \end{aligned}$$\end{document}

The error of replacing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\log Q)^2$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\log q)^2$$\end{document} is easily seen to be negligible compared to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(\log Q)^{1/2}$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Lemma 11

We now deduce the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ll 1/(\log Q)^{1-\varepsilon }$$\end{document} in (16). Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} . Applying the main result [4, Theorem 3.1] of Bettin and Drappeau to, in their notation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (x)=\lfloor 1/x \rfloor ^2$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _0=1/2-\varepsilon $$\end{document} , we conclude that there exist constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0,\delta >0$$\end{document} such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|t| < t_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{|\mathcal {F}_Q|} \sum _{p/q \in \mathcal {F}_Q} \exp \left( it \sum _{k=1}^r a_k^2 \right) = \exp \left( U(t) \log Q + O \left( |t|^{1/2-\varepsilon } + Q^{-\delta } \right) \right) , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U(t) = \frac{12}{\pi ^2} \int _{0}^{1} \frac{e^{it\lfloor 1/x \rfloor ^2} -1}{1+x} \, \textrm{d}x + O \left( |t|^{1-\varepsilon } \right) = \frac{12}{\pi ^2} \int _{1}^{\infty } \frac{e^{it \lfloor x \rfloor ^2}-1}{x^2+x} \, \textrm{d}x + O \left( |t|^{1-\varepsilon } \right) . \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0, \delta $$\end{document} and the implied constants depend only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} .

Our improvement in (16) comes from a more careful estimate for U(t). Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<t<t_0$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\lfloor x \rfloor ^2 -x^2| \le 2x$$\end{document} , the error of removing the integer part function is negligible:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \int _{1}^{\infty } \frac{e^{i t \lfloor x \rfloor ^2} - e^{i t x^2}}{x^2+x} \, \textrm{d}x \right| \le \int _{1}^{\infty } \frac{\min \{ 2 t x, 2 \}}{x^2+x} \, \textrm{d} x \ll t \log \frac{1}{t}. \end{aligned}$$\end{document}

Therefore

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U(t) = \frac{12}{\pi ^2} \int _{1}^{\infty } \frac{e^{itx^2}-1}{x^2+x} \, \textrm{d} x +O(t^{1-\varepsilon }) = \frac{12 \sqrt{t}}{\pi ^2} \int _{\sqrt{t}}^{\infty } \frac{e^{ix^2}-1}{x^2+\sqrt{t}x} \, \textrm{d} x +O(t^{1-\varepsilon }). \end{aligned}$$\end{document}

We now compare the remaining integral to its limit, the Fresnel-type integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{0}^{\infty } (e^{ix^2}-1)/x^2 \, \textrm{d} x = (i-1) \sqrt{2 \pi }/2$$\end{document} . We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \left| \int _{\sqrt{t}}^{\infty } \frac{e^{ix^2}-1}{x^2+\sqrt{t}x} \, \textrm{d} x - \int _{0}^{\infty } \frac{e^{ix^2}-1}{x^2} \, \textrm{d} x \right|&\le \left| \int _{0}^{\sqrt{t}} \frac{e^{ix^2}-1}{x^2} \, \textrm{d} x \right| \\&\quad + \int _{\sqrt{t}}^{\infty } |e^{ix^2}-1| \cdot \left| \frac{1}{x^2+\sqrt{t}x} - \frac{1}{x^2} \right| \, \textrm{d} x \\&\le \int _{0}^{\sqrt{t}} 1 \, \textrm{d} x + \int _{\sqrt{t}}^{\infty } \min \{ x^2, 2 \} \frac{\sqrt{t}}{x^3} \, \textrm{d} x \\ {}&\ll \sqrt{t} \log \frac{1}{t}, \end{aligned} \end{aligned}$$\end{document}

hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(t)= \frac{6 \sqrt{2} \sqrt{t}}{\pi ^{3/2}} (i-1) + O(t^{1-\varepsilon })$$\end{document} . The case of negative t follows from complex conjugation, thus for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|t|<t_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U(t) = - \frac{6 \sqrt{2} |t|^{1/2}}{\pi ^{3/2}} (1- i \textrm{sgn}(t)) + O(|t|^{1-\varepsilon }) . \end{aligned}$$\end{document}

Now let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varphi _1(t)= \frac{1}{|\mathcal {F}_Q|} \sum _{p/q \in \mathcal {F}_Q} \exp \left( i t \frac{\pi ^3}{72 (\log Q)^2} \sum _{k=1}^r a_k^2 \right) \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _2(t)=\exp (-|t|^{1/2}(1-i \textrm{sgn}(t)))$$\end{document} ; the latter is the characteristic function of the standard Lévy distribution. The Berry–Esseen inequality [24, p. 142] states that the distance of these two distributions in the Kolmogorov metric is, with any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} \sup _{t \ge 0} \left| \frac{1}{|\mathcal {F}_Q|} \left| \left\{ \frac{p}{q} \in \mathcal {F}_Q \,: \, \frac{\pi ^3}{72 (\log Q)^2} \sum _{k=1}^r a_k^2 \le t \right\} \right| - \int _0^t \frac{e^{-1/(2x)}}{\sqrt{2 \pi } x^{3/2}} \, \textrm{d} x \right| \\ {}{} & {} \qquad \ll \frac{1}{T} + \int _{0}^{T} \frac{|\varphi _1(t)-\varphi _2(t)|}{t} \, \textrm{d} t. \end{aligned}$$\end{document}

Choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = \log Q$$\end{document} . Formulas (19) and (20) show that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|t| \le \log Q$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \varphi _1(t)&= \varphi _2(t) \exp \left( O \left( \left( \frac{|t|}{(\log Q)^2} \right) ^{1-\varepsilon } \log Q + \left( \frac{|t|}{(\log Q)^2} \right) ^{1/2-\varepsilon } + Q^{-\delta } \right) \right) \\ {}&= \varphi _2(t) \left( 1 + O \left( \frac{|t|^{1-\varepsilon }+|t|^{1/2-\varepsilon }}{(\log Q)^{1-2 \varepsilon }} + Q^{-\delta } \right) \right) . \end{aligned} \end{aligned}$$\end{document}

Using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varphi _2(t)|= e^{-|t|^{1/2}}$$\end{document} , this immediately yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\varphi _1(t)-\varphi _2(t)| \ll e^{-|t|^{1/2}} \left( \frac{|t|^{1-\varepsilon }+|t|^{1/2-\varepsilon }}{(\log Q)^{1-2 \varepsilon }} + Q^{-\delta } \right) . \end{aligned}$$\end{document}

It is now easy to see that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{Q^{-100}}^{1} \frac{|\varphi _1(t)-\varphi _2(t)|}{t} \, \textrm{d} t \ll \frac{1}{(\log Q)^{1-2 \varepsilon }} \quad \text {and} \quad \int _{1}^{\log Q} \frac{|\varphi _1(t)-\varphi _2(t)|}{t} \, \textrm{d} t \ll \frac{1}{(\log Q)^{1-2 \varepsilon }}. \end{aligned}$$\end{document}

On the other hand, by a very rough estimate we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^r a_k^2 \le Q^3$$\end{document} , hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varphi _1(t)-1| \ll |t| Q^3$$\end{document} . Clearly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varphi _2(t)-1| \ll |t|^{1/2}$$\end{document} , thus

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{0}^{Q^{-100}} \frac{|\varphi _1(t)-\varphi _2(t)|}{t} \, \textrm{d} t \ll \int _{0}^{Q^{-100}} \frac{t Q^3 + t^{1/2}}{t} \, \textrm{d} t \ll Q^{-50}. \end{aligned}$$\end{document}

Therefore

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{\log Q} + \int _{0}^{\log Q} \frac{|\varphi _1(t)-\varphi _2(t)|}{t} \, \textrm{d} t \ll \frac{1}{(\log Q)^{1-2 \varepsilon }}, \end{aligned}$$\end{document}

as claimed.