Hausdorff dimension of the large values of Weyl sums

TL;DR

This paper investigates the Hausdorff dimension of sets of vectors with large Weyl sums, providing an upper bound that complements previous lower bounds, thus advancing understanding of their fractal structure.

Contribution

It introduces an upper bound for the Hausdorff dimension of vectors with large Weyl sums, complementing earlier lower bounds and refining the fractal analysis of these sets.

Findings

Established an upper bound for the Hausdorff dimension of large Weyl sum sets.

Complemented previous lower bounds, narrowing the possible dimension range.

Enhanced understanding of the fractal geometry of Weyl sum exceptional sets.

Abstract

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| \ge N^{\alpha} $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

Equations

$$\left|\sum _{n=1}^N\exp \left(2\pi i\left(x_{1}n+\dots +x_d{n}^d\right)\right)\right|⩾N^{\alpha }$$

$$S_d(\mathbf{x};N)=\sum _{n=1}^N\mathbf{e}\left(x_{1}n+\dots +x_d{n}^d\right),$$

$$|S_d(\mathbf{x};N)|⩽N^{1/2+o(1)}\text{ as }N\to \infty ,$$

$$\underset{N\to \infty }{\mathrm{limsup}}\left|S_2((x_1,x_2);N)\right|N^{-1/2}(\log \log N{)}^{-1/6}=\infty .$$

$${\mathcal{E}}_{d,\alpha }=\{\mathbf{x}\in {\mathsf{T}}_d:\text{leavevmode}|S_d(\mathbf{x};N)|⩾N^{\alpha }\text{ for infinitely many }N\in \mathbb{N}\}.$$

$$\begin{split}\Xi_{d}=\bigl{\{}\mathbf{x}\in\mathsf{T}_{d}:\leavevmode\nobreak\ \forall\varepsilon>0,|S_{d}(\mathbf{x};N)|&\geqslant N^{1-\varepsilon}\\ &\text{for infinitely many }N\in{\mathbb{N}}\bigr{\}}\end{split}$$

$$\dim {\mathcal{E}}_{d,\alpha }⩾\xi (d,\alpha )>0$$

$${\mathcal{H}}^s(\mathcal{A})=\underset{\delta \to 0}{\lim }{\mathcal{H}}_{\delta }^s(\mathcal{A}),$$

$${\mathcal{H}}_{\delta }^s(\mathcal{A})=\inf \left\{\sum _{i=1}^{\infty }{\left(\mathrm{diam}{\mathcal{U}}_i\right)}^s:\text{leavevmode}\mathcal{A}\subseteq \bigcup _{i=1}^{\infty }{\mathcal{U}}_i\text{ and }\mathrm{diam}{\mathcal{U}}_i⩽\delta ,i\in \mathbb{N}\right\}.$$

$$\dim \mathcal{A}$$

$$=\sup \{s>0:\text{leavevmode}{\mathcal{H}}^s(\mathcal{A})=\infty \}.$$

$$\mathfrak{u}(d,\alpha )=\underset{k=0,\dots ,d-1}{\min }\frac{(2d^2+4d)(1-\alpha )+k(k+1)}{4-2\alpha +2k}.$$

$$\dim {\mathcal{E}}_{d,\alpha }⩽\mathfrak{u}(d,\alpha ).$$

$$\mathfrak{u}(d,\alpha )⩽d-\frac{d(d+1)(2\alpha -1)}{2(d+1-\alpha )}.$$

$$\dim {\mathcal{E}}_{d,\alpha }⩽\mathfrak{u}(d,\alpha )⩽\frac{(2d^2+4d)(1-\alpha )}{4-2\alpha }.$$

$$c_1(d)⩽\underset{\alpha \to 1}{\mathrm{liminf}}(1-\alpha {)}^{-1}\dim {\mathcal{E}}_{d,\alpha }⩽\underset{\alpha \to 1}{\mathrm{limsup}}(1-\alpha {)}^{-1}\dim {\mathcal{E}}_{d,\alpha }⩽c_2(d)$$

$$c_1(2)=3\text{mbox}and{c}_2(2)=8,$$

$$c_1(d)=\underset{\nu =1,\dots ,d}{\max }\min \left\{\frac{1}{\nu },\frac{2}{2d-\nu }\right\}\text{mbox}and{c}_2(d)=d^2+2d.$$

$${\int }_{{\mathsf{T}}_d}|S_d(\mathbf{x};N){|}^{2s(d)}d\mathbf{x}⩽N^{s(d)+o(1)},$$

$${\int }_{{\mathsf{T}}_d}|\sum _{n=1}^N{a}_n\mathbf{e}(x_{1}n+\dots +x_d{n}^d){|}^{2s(d)}d\mathbf{x}⩽N^{o(1)}{\left(\sum _{n=1}^N|a_n{|}^2\right)}^{s(d)}.$$

$$S_d(\mathbf{x};M)\ll W_d(\mathbf{x};N),$$

$$W_d(\mathbf{x};N)=\sum _{h=1}^N\frac{1}{h}\left|\sum _{n=1}^N\mathbf{e}(hn/N)\mathbf{e}\left(x_{1}n+\dots +x_d{n}^d\right)\right|.$$

$$f(n)=x_{1}n+\dots +x_d{n}^d.$$

$$\frac{1}{N}\sum _{h=1}^N\sum _{k=1}^M\mathbf{e}\left(h(n-k)/N\right)=\{\begin{array}{ll}1& n=1,\dots ,M,\\ 0& \text{otherwise}.\end{array}$$

$$\sum _{k=1}^M\mathbf{e}\left(hk/N\right)\ll \frac{N}{\min \{h,N+1-h\}},$$

$$S_d(\mathbf{x};M)$$

$$=\frac{1}{N}\sum _{h=1}^N\sum _{k=1}^M\mathbf{e}\left(-hk/N\right)\sum _{n=1}^N\mathbf{e}(f(n))\mathbf{e}\left(hn/N\right)$$

$$\ll \sum _{h=1}^N\frac{1}{\min \{h,N+1-h\}}\left|\sum _{n=1}^N\mathbf{e}\left(hn/N\right)\mathbf{e}(f(n))\right|$$

$$\ll \sum _{h=1}^N\frac{1}{h}\left|\sum _{n=1}^N\mathbf{e}\left(hn/N\right)\mathbf{e}(f(n))\right|,$$

$$b_N(n)\ll \log N,n=1,\dots ,N,$$

$$W_d(\mathbf{x};N)=\sum _{n=1}^N{b}_N(n)\mathbf{e}(x_{1}n+\dots +x_d{n}^d).$$

$${\mathcal{E}}_{d,\alpha +\eta }\subseteq \{\mathbf{x}\in {\mathsf{T}}_d:\text{leavevmode}|W_d(\mathbf{x};N_i)|⩾N_i^{\alpha }\text{ for infinitely many }i\in \mathbb{N}\}.$$

Full text

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for infinitely many integers $N\geqslant 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

1. Introduction

1.1. Motivation and background

For an integer $d\geqslant 2$, let $\mathsf{T}_{d}=({\mathbb{R}}/{\mathbb{Z}})^{d}$ be the $d$-dimensional unit torus.

For a vector $\mathbf{x}=(x_{1},\ldots,x_{d})\in\mathsf{T}_{d}$ and $N\in{\mathbb{N}}$, we consider the exponential sums

[TABLE]

which are commonly called Weyl sums, where throughout the paper we denote ${\mathbf{\,e}}(x)=\exp(2\pi ix)$.

The authors [2, Appendix A] have shown that for almost all $\mathbf{x}\in\mathsf{T}_{d}$ (with respect to Lebesgue measure) one has

[TABLE]

see also [3, Theorem 2.1] for a different proof. It is very natural to conjecture that the exponent $1/2$ is the best possible value, and indeed for $d=2$ the authors [4] proved that for almost all $(x_{1},x_{2})\in\mathsf{T}_{2}$ we have

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However there seems to be no results in this direction for $d\geqslant 3$.

For integer $d\geqslant 2$ and $0<\alpha<1$ our main object is defined as

[TABLE]

We can restate the bound (1.1) in the following way: for any $\alpha\in(1/2,1)$ the set ${\mathcal{E}}_{d,\alpha}$ is of Lebesgue measure zero. Here we are mostly interested in the structure of the sets ${\mathcal{E}}_{d,\alpha}$, and for convenience we call the set ${\mathcal{E}}_{d,\alpha}$ the exceptional set for any integer $d\geqslant 2$ and each $0<\alpha<1$.

The authors [2] show that in terms of the Baire categories and Hausdorff dimension the exceptional sets ${\mathcal{E}}_{d,\alpha}$ are quite massive. By [2, Theorem 1.3], for each $0<\alpha<1$ and integer $d\geqslant 2$ the set $\mathsf{T}_{d}\setminus{\mathcal{E}}_{d,\alpha}$ is of the first Baire category. Alternatively, this is equivalent to the statement that the complement $\mathsf{T}_{d}\setminus\Xi_{d}$ to the set

[TABLE]

is of first category, see [2] for more details and reference therein. For the Hausdorff dimension it is shown in [2, Theorem 1.5] that for any $d\geqslant 2$ and $0<\alpha<1$ one has

[TABLE]

with some explicit constant $\xi(d,\alpha)$.

We remark that the authors [3, Corollary 1.9] have obtained a nontrivial upper bound for the Hausdorff dimension of ${\mathcal{E}}_{d,\alpha}$ for some $\alpha$, however the bounds there are not fully explicit and do not cover the whole range $1/2<\alpha<1$.

Here we obtain the nontrivial upper bound of $\dim{\mathcal{E}}_{d,\alpha}$ for all $1/2<\alpha<1$ and $d\geqslant 2$.

On the other hand, we note that we do not have any plausible conjecture about the exact value of the Hausdorff dimension of ${\mathcal{E}}_{d,\alpha}$.

1.2. Main results

For ${\mathcal{A}}\subseteq{\mathbb{R}}^{d}$, the $s$-dimension Hausdorff measure of ${\mathcal{A}}$ is defined as

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where

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The Hausdorff dimension of ${\mathcal{A}}$ is defined as

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We refer to [5] for more details and properties of Hausdorff dimension.

For integer $d\geqslant 2$ and $0<\alpha<1$ denote

[TABLE]

Theorem 1.1**.**

For any integer $d\geqslant 2$ and $0<\alpha<1$ we have

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For $d\geqslant 2$ and any $1/2<\alpha<1$ an elementary calculation gives that $\mathfrak{u}(d,\alpha)<d$. In fact by taking $k=d-1$ in (1.4) we derive

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Thus, we have

Corollary 1.2**.**

For any integer $d\geqslant 2$ and any $1/2<\alpha<1$ we have $\dim{\mathcal{E}}_{d,\alpha}<d.$

Furthermore taking, for example, $k=0$ in (1.4) we obtain

[TABLE]

We note that although the lower bound (1.3) and the upper bound of Theorem 1.1 are of very different magnitude with respect to $d$, however for $\alpha\to 1$ they give the same rate of convergency to zero which of order $1-\alpha$. More precisely, the explicit formula for $\xi(d,\alpha)$ from [2] and the formula (1.4) yield

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for two positive constants $c_{1}(d),c_{2}(d)$ depending only on $d$. In fact for $d=2$ we have

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while for $d\geqslant 3$ we have

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In particular, we have

Corollary 1.3**.**

For any integer $d\geqslant 2$, if $\alpha\to 1$ then $\dim{\mathcal{E}}_{d,\alpha}\rightarrow 0$.

From the definition of $\Xi_{d}$, see (1.2), we have $\Xi_{d}\subseteq{\mathcal{E}}_{d,\alpha}$ for any $0<\alpha<1$. Therefore

Corollary 1.4**.**

For and integer $d\geqslant 2$, we have $\dim\Xi_{d}=0$.

2. Preliminaries

2.1. Notation and conventions

Throughout the paper, the notation $U=O(V)$, $U\ll V$ and $V\gg U$ are equivalent to $|U|\leqslant c|V|$ for some positive constant $c$, which throughout the paper may depend on the degree $d$ and occasionally on the small real positive parameters $\varepsilon$ and $\delta$.

For any quantity $V>1$ we write $U=V^{o(1)}$ (as $V\to\infty$) to indicate a function of $V$ which satisfies $|U|\leqslant V^{\varepsilon}$ for any $\varepsilon>0$, provided $V$ that is large enough.

We use $\#{\mathcal{X}}$ to denote the cardinality of set ${\mathcal{X}}$.

We always identify $\mathsf{T}_{d}$ with half-open unit cube $[0,1)^{d}$, in particular we naturally associate Euclidean norm $\|x\|$ with points $x\in\mathsf{T}_{d}$. Moreover we always assume that $d\geqslant 2$.

We say that some property holds for almost all $\mathbf{x}\in\mathsf{T}_{d}$ if it holds for a set ${\mathcal{X}}\subseteq\mathsf{T}_{d}$ of Lebesgue measure $\lambda({\mathcal{X}})=1$.

We always keep the subscript $d$ in notations for our main objects of interest such as ${\mathcal{E}}_{d,\alpha}$, $S_{d}(\mathbf{x};N)$ and $\mathsf{T}_{d}$, but sometimes suppress it in auxiliary quantities.

2.2. Mean value theorems

The Vinogradov mean value theorem in the currently known form, due to Bourgain, Demeter and Guth [1] for $d\geqslant 4$ and Wooley [7] for $d=3$, asserts that,

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where $s(d)=d(d+1)/2$. We will use the following result due to Wooley [9, Theorem 1.1], which extends the bound to the Weyl sums with weights.

Lemma 2.1**.**

For any sequence of complex weights $\mathbf{a}=(a_{n})_{n=1}^{\infty}$, and any integer $N\geqslant 1$, we have the upper bound

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2.3. Completion method

The following bound is special case of [3, Lemma 3.2], and for completeness we give a proof here.

Lemma 2.2**.**

For $\mathbf{x}\in\mathsf{T}_{d}$ and $1\leqslant M\leqslant N$ we have

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where

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Proof.

For $\mathbf{x}\in\mathsf{T}_{d}$ and $n\in{\mathbb{N}}$ denote

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Observe that by the orthogonality

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We also note that for $1\leqslant h,M\leqslant N$ we have

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see [6, Equation (8.6)]. It follows that

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which finishes the proof.     \sqcap$$\sqcup

Observe that for any $N$ there exists a sequence $b_{N}(n),n=1,\ldots,N$ such that

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and $W_{d}(\mathbf{x};N)$ can be written as

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From Lemma 2.2 we immediately obtain:

Corollary 2.3**.**

Let $0<\alpha<1$ and $N_{i}=2^{i},i\in{\mathbb{N}}$. Using above notation for any $\eta>0$ we have

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Thus for the purpose of estimate the set ${\mathcal{E}}_{d,\alpha}$ it is sufficient to know the size of the set

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which we investigate in Section 2.4 below.

2.4. Distribution of large values of exponential sums

We first remark that the results in this subsection are special forms of [3], see also [8, Lemma 2.1]. For completeness we give proofs for these special cases.

For $\mathbf{u}\in{\mathbb{R}}^{d}$ and $\bm{\zeta}=(\zeta_{1},\ldots,\zeta_{d})$ with $\zeta_{j}>0$, $j=1,\ldots,d$, we define the $d$-dimensional rectangle (or box) with the centre $\mathbf{u}$ and the side lengths $2\bm{\zeta}$ by

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In analogue of [8, Lemma 2.1] and [3, Lemma 3.5] we obtain:

Lemma 2.4**.**

Let $0<\alpha<1$ and let $\varepsilon>0$ be sufficiently small. If $|W_{d}(\mathbf{x};N)|\geqslant N^{\alpha}$ for some $\mathbf{x}\in\mathsf{T}_{d}$, then

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holds for any $\mathbf{y}\in{\mathcal{R}}(\mathbf{x},\bm{\zeta})$ provided that $N$ is large enough and

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Proof.

For any $h=1,\ldots,N$ we have

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The last estimate holds for all large enough $N$. By Lemma 2.2 we obtain

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which holds for all large enough $N$ and gives the result.     \sqcap$$\sqcup

In analogue of [3, Lemma 3.7] from Lemmas 2.2 and 2.4 we obtain:

Lemma 2.5**.**

Let $0<\alpha<1$ and $\varepsilon>0$ be a small parameter. For each $j=1,\ldots,d$ let

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We divide $\mathsf{T}_{d}$ into

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boxes of the type

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where $n_{j}=0,\ldots,1/\zeta_{j}-1$, $j=1,\ldots,d$. Let $\mathfrak{R}$ be the collection of these boxes, and

[TABLE]

Then one has

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Proof.

Let ${\mathcal{R}}\in\mathfrak{R}$. By Lemma 2.4 if $|W_{d}(\mathbf{x};N)|\geqslant N^{\alpha}$ for some $\mathbf{x}\in{\mathcal{R}}$, then for any $\mathbf{y}\in{\mathcal{R}}$ we have $|W_{d}(\mathbf{y};N)|\geqslant N^{\alpha}/2$. Combining with Lemma 2.1 and (2.1),  (2.2) we derive

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which yields the desired bound.     \sqcap$$\sqcup

Note that the above bound of $\#\widetilde{\mathfrak{R}}$ is nontrivial when $1/2<\alpha<1$.

From Corollary 2.3 and Lemma 2.5 we formulate the following Corollary 2.6 for the convenience of our applications.

Corollary 2.6**.**

Let $0<\alpha<1$ and $N_{i}=2^{i},i\in{\mathbb{N}}$. Then for any $\eta>0$ we have

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where each ${\mathcal{R}}$ of $\mathfrak{R}(i)$ has the side length $\bm{\zeta}=(\zeta_{1},\ldots,\zeta_{d})$ such that

[TABLE]

and furthermore

[TABLE]

3. Proof of Theorem 1.1

We start from some auxiliary results. First, we adapt the definition of the singular value function from [5, Chapter 9] to the following.

Definition 3.1**.**

Let ${\mathcal{R}}\subseteq{\mathbb{R}}^{d}$ be a rectangle with side lengths

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For $0<t\leqslant d$ we set

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and for $k=1,\ldots,d-1$ we define

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Note that for a rectangle ${\mathcal{R}}\subseteq{\mathbb{R}}^{2}$ with the side length $r_{1}\geqslant r_{2}$ we have

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Remark 3.2**.**

The notation $\varphi_{k,t}({\mathcal{R}})$ roughly means that we can cover the rectangle ${\mathcal{R}}$ by about (up to a constant factor)

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balls of radius $r_{k+1}$, and hence this leads to the term

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in the expression for the Hausdorff measure with the parameter $t$ (again up to a constant factor which does not affect our results).

From the definition of the Hausdorff dimension, using the above notation, we have the following inequality

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Now we turn to the proof of Theorem 1.1. For $k=1,\ldots,d-1$ and $0<t\leqslant d$ we have

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Here and in the following we denote

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We remark that (3.2) also holds for the case $k=0$, in which we have $s(k)=0$. To be precise for $k=0$ we have

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Applying (3.1) we conclude that

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provided that the parameters $\alpha,\rho,k,t$ satisfy the following further condition

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which becomes

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By the arbitrary choice of $\eta$ we finish the proof.

Acknowledgement

This work was supported by ARC Grant DP170100786.