Hardy-Littlewood series and even continued fractions

TL;DR

This paper characterizes the set of points where the Hardy-Littlewood series converges on [-1,1], using even continued fractions, and provides an intrinsic description of convergence sets based on measure theory.

Contribution

It introduces a new approach to describe convergence sets of the Hardy-Littlewood series using even continued fractions and an approximate functional equation.

Findings

Identifies full measure subsets where the series converges.

Establishes convergence of series related to even continued fraction convergents.

Provides an intrinsic measure-theoretic description of convergence sets.

Abstract

For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{i\pi n^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for $F_s$. In this paper, we define in terms of even or regular continued fractions certain subsets of points of $[-1,1]$ of full measure where the series converges. Our method is based on an approximate function equation for $F_s(x)$. As a by-product, we obtain the convergence of certain series defined in term of the convergents of the even continued fraction of an irrational number.