Small values of Weyl sums

TL;DR

This paper investigates the behavior of Weyl sums and generalized Gaussian sums, showing that the set of points where these sums tend to zero is dense, contains a large subset, and has positive Hausdorff dimension.

Contribution

It establishes that the set of points with vanishing Weyl sums is dense, large in Hausdorff dimension, and extends results to generalized Gaussian sums.

Findings

The set of points with Weyl sums tending to zero is dense in [0,1)^d.

This set has a positive Hausdorff dimension.

Similar properties are proven for generalized Gaussian sums.

Abstract

We prove that the set of $(x_1, \ldots, x_d)\in [0,1)^d$, such that $$ \underline{\lim}_{N\to \infty}\left| \sum_{n=1}^N\exp(2 \pi i (x_1n+\ldots + x_dn^d)) \right| =0, $$ contains a dense $\mathcal{G}_\delta$ set in $[0,1)^d$ and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums $$ \sum_{n=1}^N\exp(2\pi i x n^d), \qquad x \in [0,1). $$

Equations

$$\underset{N\to \infty }{\underline{\lim }}\left|\sum _{n=1}^N\exp (2\pi i(x_{1}n+\dots +x_d{n}^d))\right|=0,$$

$$\sum _{n=1}^N\exp (2\pi ix{n}^d),x\in [0,1).$$

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

$$G_d(x;N)=\sum _{n=1}^{N}e(x{n}^d),$$

$$\mathsf{T}={\mathsf{T}}_1=\mathbb{R}/\mathbb{Z}.$$

$$G(x;N)=G_2(x;N).$$

$$\left|G(x;N)\right|⩾c\sqrt{N}$$

$$c\sqrt{N}⩽\left|S_2\left((y,x);N\right)\right|⩽C\sqrt{N}.$$

$$\underset{N\to \infty }{\overline{\lim }}\frac{\left|G(x;N)\right|}{\sqrt{N}f(N)}<\infty ⟺\sum _{n=1}^{\infty }\frac{1}{nf(n{)}^4}<\infty .$$

$${\Psi }_N(\alpha )=\lambda (\{x\in [0,1]:N^{-1/2}|\sum _{n=1}^N\mathbf{e}(n^{2}x)|⩾\alpha \}),$$

$$N^{-1/2}\sum _{n=1}^N\mathbf{e}(n^{2}x)$$

$$\underset{N\to \infty }{\overline{\lim }}\frac{\left|S_2(\mathbf{x};N)\right|}{\sqrt{N}g(\ln N)}<\infty ⟺\sum _{n=1}^{\infty }\frac{1}{g(n{)}^6}<\infty .$$

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

$$\mathcal{B}(x)=\{y\in \mathsf{T}:\text{leavevmode}{S}_2\left((y,x);N\right),N\in \mathbb{N},\text{ is dense in }\mathbb{C}\}.$$

$$\underset{q\to \infty }{\underline{\lim }}{q}^{3/2}\Vert qx\Vert <\infty ,$$

$$\underset{q\to \infty }{\underline{\lim }}{q}^{3+\epsilon }\Vert qx\Vert =0,$$

$$\underset{q\to \infty }{\underline{\lim }}{q}^d\Vert q{x}_d\Vert <\infty ,$$

$$S_d(\mathbf{x};N),N=1,2,\dots ,$$

$$\underset{q\to \infty }{\underline{\lim }}{q}^{d-1/2}\Vert q{x}_d\Vert <\infty ,$$

$$S_d(\mathbf{x};N),N=1,2,\dots ,$$

$${\mathcal{Z}}_{d,W}=\{\mathbf{x}\in {\mathsf{T}}_d:\text{leavevmode}\underset{N\to \infty }{\underline{\lim }}|S_d(\mathbf{x};N)|=0\},$$

$${\mathcal{Z}}_{d,G}=\{x\in \mathsf{T}:\text{leavevmode}\underset{N\to \infty }{\underline{\lim }}|G_d(x;N)|=0\}.$$

$$\displaystyle\Xi_{d}=\bigl{\{}\mathbf{x}\in\mathsf{T}_{d}:\leavevmode\nobreak\ \forall\varepsilon>0,\,\operatorname*{\overline{lim}}_{N\to\infty}\frac{|S_{d}(\mathbf{x};N)|}{N^{1-\varepsilon}}=\infty\bigr{\}}.$$

$$\underset{N\to \infty }{\underline{\lim }}|S_d(\mathbf{x};N)|=0\text{mbox}and\underset{N\to \infty }{\underline{\lim }}|G_d(x;N)|=0,$$

$$\underset{N\to \infty }{\overline{\lim }}\frac{|S_d(\mathbf{x};N)|}{N^{1-\epsilon }}=\infty \text{mbox}and\underset{N\to \infty }{\overline{\lim }}\frac{|G_d(x;N)|}{N^{1-\epsilon }}=\infty ,$$

$$\displaystyle\dim_{H}{\mathcal{A}}=\inf\Bigl{\{}s>0:\leavevmode\nobreak\ \forall\,\varepsilon>0,$$

$$\displaystyle{\mathcal{A}}\subseteq\bigcup_{i=1}^{\infty}{\mathcal{U}}_{i}\text{ and }\sum_{i=1}^{\infty}\left(\operatorname{diam}{\mathcal{U}}_{i}\right)^{s}<\varepsilon\Bigr{\}}.$$

$${\dim }_H{\mathcal{Z}}_{d,W}⩾6/5\text{mbox}and{\dim }_H{\mathcal{Z}}_{d,G}⩾4/(2d+1).$$

$${\mathbf{e}}_m(z)=\mathbf{e}(z/m).$$

$$\underset{1⩽N⩽p}{\max }\left|\sum _{n=1}^N{\mathbf{e}}_p(b{n}^2)\right|\ll \sqrt{p}.$$

$$\underset{1⩽M,N⩽p}{\max }\left|\sum _{M+1⩽n⩽M+N}{\mathbf{e}}_p\left(an+b{n}^2\right)\right|\ll \sqrt{p}.$$

Full text

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contains a dense ${\mathcal{G}}_{\delta}$ set in $[0,1)^{d}$ and has positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums

[TABLE]

1. Introduction

1.1. Motivation

For an integer $d\geqslant 2$, let $\mathsf{T}_{d}=({\mathbb{R}}/{\mathbb{Z}})^{d}$ denote 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 Weyl sums

[TABLE]

where throughout the paper we denote ${\mathbf{\,e}}(x)=\exp(2\pi ix)$. Since we also consider monomial sums

[TABLE]

it is also convenient to define

[TABLE]

We recall that for $d=2$ the sums $G_{2}(x;N)$ are also known as Gaussian sums and we denote

[TABLE]

Obtaining upper bounds on these sums has received a lot attention over the last several decades, most significantly due the the proof of the called main conjecture in the Vinogradov mean value theorem by Bourgain, Demeter and Guth [5] (for $d\geqslant 4$) and Wooley [40] (for $d=3$) (see also [41]). In particular, the state of art is conveniently summarised by Bourgain [4, Theorem 5].

On the other hand, the problem of the distribution of values and in particular of lower bounds seems to be less know with just several sporadic results in the literature, mostly related to the case $d=2$, and those do not seem to be widely known. Motivated by this, we first give a survey of known results about the measure and topological properties of sets of $\mathbf{x}\in\mathsf{T}_{d}$ and $x\in\mathsf{T}$ large sums $S_{d}(\mathbf{x};N)$ and $G_{d}(x;N)$ and then obtain two new results.

1.2. A survey of results on the distribution of Weyl sums

We first show some known results for the case $d=2$. The study of the sums $S_{2}(\mathbf{x};N)$ and $G(x;N)$ has been initiated by Hardy and Littlewood [23].

First we recall we say that $x\in T$ has bounded partial quotients if $\sup_{n\in{\mathbb{N}}}a_{n}<\infty$, where $x=[a_{1},a_{2},\ldots]$ is the continued fraction representation of $x$. Hardy and Littlewood [23] have given the following lower and upper bounds.

Theorem 1.1**.**

(Hardy and Littlewood [23, Theorems 2.22 and 2.25])*

Let $x\in\mathsf{T}$ be irrational. Then*

• (i)

there exists a constant $c>0$ such that

[TABLE]

for infinitely many $N\in{\mathbb{N}}$;

• (ii)

if the continued fraction of $x$ has bounded partial quotients, then there exist absolute constants $C>c>0$ such that for any $y\in\mathsf{T}$ one has

[TABLE]

Furthermore, except for a set of $x\in\mathsf{T}$ of Lebesgue measure zero, Fiedler, Jurkat and Körner [19, Theorem 2] have obtained the optimal bound for the sums $G(x;N)$.

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 $\lambda({\mathcal{X}})=1$.

Theorem 1.2**.**

(Fiedler, Jurkat and Körner [19, Theorem 2])*

Suppose that $\{f(n)\}_{n=1}^{\infty}$ is a non-decreasing sequence of positive numbers. Then for almost all $x\in\mathsf{T}$ one has*

[TABLE]

For $x\in[0,1]$ instead of obtaining the bound of the sums $\sum_{n=1}^{N}{\mathbf{\,e}}(n^{2}x)$, Jurkat and van Horne [26, 27, 28] studied the sequence of the distribution function

[TABLE]

where $\lambda$ is the Lebesgue measure. Among other things Jurkat and van Horne [26, 27, 28] proved that $\Psi_{N}$ convergences to a limiting distribution (the limit is not normal distribution).

Marklof [35] has studied the asymptotic behaviour of the sums

[TABLE]

in the complex plane ${\mathbb{C}}$ as $N\rightarrow\infty$ for all $x\in{\mathbb{R}}$.

We remark that the methods of Jurkat and van Horne [26, 27, 28] are mainly based on the circle method, Diophantine approximations, and bounds of Kloosterman sums, while the approach of Marklof [35] stems from the theory of dynamical systems.

For quadratic Weyl sums, a generalisation of Theorem 1.2 is given by Fedotov and Klopp [18, Theorem 0.2]). Furthermore, Fedotov and Klopp [18] have given a similar result for the sums $S_{2}(\mathbf{x};N)$, however adding the term ${\mathbf{\,e}}(x_{1}n)$ leads to more cancellations in the sums $S_{2}(\mathbf{x};N)$.

Theorem 1.3**.**

(Fedotov and Klopp [18, Theorem 0.1])*

Suppose that $\{g(n)\}_{n=1}^{\infty}$ is a non-decreasing sequence of positive numbers. Then for almost all $\mathbf{x}\in\mathsf{T}_{2}$ one has*

[TABLE]

For the case $d\geqslant 3$, the authors in [10, Appendix A], and [11, Theorem 2.1] have shown in two different ways that for almost all $\mathbf{x}\in\mathsf{T}_{d}$ one has

[TABLE]

It is natural to conjecture that the bound (1.1) is close to the best possible up to the $N^{o(1)}$ term, see also [10, Conjecture 1.1]. We remark that the conjecture is still open for the case $d\geqslant 3$.

Observe that we may consider the sums $S_{d}(\mathbf{x};N)$, $N\in{\mathbb{N}}$, as a sequence of points in the complex plane $\mathbb{C}$. Before giving some results in this direction we show some notation first.

We now need to recall some standard definitions.

Definition 1.4**.**

We sat that a set ${\mathcal{S}}$ of a topological space ${\mathcal{X}}$ is a ${\mathcal{G}}_{\delta}$-set if it is a countable intersection of open sets.

We could say that a dense ${\mathcal{G}}_{\delta}$ set is a “large” set in the sense of topology. We remark that the ${\mathcal{G}}_{\delta}$ set is closely related to Baire categories, see [37, Section 9] for more details.

Given $x\in\mathsf{T}$ define

[TABLE]

For a real $x$ the notation we use $\|x\|$ to denote the distance to the closest integers.

Using an approach which stems from the theory of dynamical systems and considering the Weyl sums as a cocycle on $\mathbb{C}$, Forrest [20] have obtained the following result.

Theorem 1.5**.**

(Forrest [20, Theorem 2])*

If $x\in\mathsf{T}$ is irrational and*

[TABLE]

then ${\mathcal{B}}(x)$ contains a dense ${\mathcal{G}}_{\delta}$ set in $\mathsf{T}$.

There are serval interesting generations of Theorem 1.5.

Theorem 1.6**.**

(Forrest [21, Theorem 1.3])* Suppose that $x\in\mathsf{T}$ is irrational and has continued fraction representation $[a_{1},a_{2},\ldots]$ such that $\sum_{n=1}^{\infty}1/a_{n}<\infty$, and suppose that*

[TABLE]

for some $\varepsilon>0$. Then ${\mathcal{B}}(x)$ is of full Lebesgue measure in $\mathsf{T}$.

We remark that Fayad [17] and Greschonig, Nerurkar and Volný [22] studied the ergodic property of the dynamical system which was introduced by Forrest [20, 21] (for the purpose of studying the distribution of Weyl sums). We refer to [17, 22] for more details and reference therein.

We remark that the following conjecture of Forrest [20] is still open.

Conjecture 1.7**.**

(Forrest [20])*

Theorem 1.5 is true for every irrational $x$ with $\operatorname*{\underline{lim}}_{q\to\infty}q\|qx\|=0$.*

Forrest [20] also considered the sequence $S_{d}(\mathbf{x};N),N\in{\mathbb{N}}$ for the case $d\geqslant 3$. In analogy of Theorem 1.5, Forrest [20] showed the following.

Theorem 1.8**.**

(Forrest [20, Theorem 10])* Suppose that $d\geqslant 3$ and $x_{d}\in\mathsf{T}$ is irrational such that*

[TABLE]

then for a dense $G_{\delta}$ set of $(x_{1},\ldots,x_{d-1})\in[0,1]^{d-1}$, the sequence of partial sums

[TABLE]

is dense in $\mathbb{C}$.

With the weaker condition for the leading term $x_{d}$, Forrest [20] gives the following result.

Theorem 1.9**.**

(Forrest [20, Proposition 13])*

Suppose that $d\geqslant 3$ and $x_{d}$ is irrational such that*

[TABLE]

then for a dense $G_{\delta}$ set of $(x_{1},\ldots,x_{d-1})\in\mathsf{T}_{d-1}$ the partial sums

[TABLE]

is dense at [math] in $\mathbb{C}$.

One may also conjecture that for almost all $\mathbf{x}\in\mathsf{T}_{d}$ the sequence $S_{d}(\mathbf{x};N),N\in{\mathbb{N}}$ is dense in $\mathbb{C}$, see Conjecture 4.1 below.

1.3. Main results

We now show that there are many very small quadratic Weyl sums and monomial sums for arbitrary degree. More precisely we define the “zero sets” of Weyl sums and monomial sums as

[TABLE]

and

[TABLE]

We now show that ${\mathcal{Z}}_{d,W}$ and ${\mathcal{Z}}_{d,G}$ are quite massive.

Theorem 1.10**.**

For any integer $d\geqslant 2$, the sets ${\mathcal{Z}}_{d,W}$ and ${\mathcal{Z}}_{d,G}$ contain dense ${\mathcal{G}}_{\delta}$ sets in $\mathsf{T}_{d}$ and $\mathsf{T}$, respectively.

We remark that the claim of Theorem 1.10 for ${\mathcal{Z}}_{d,W}$ can perhaps be derived from Theorem 1.9, however we prove it via a completely different method, which also applies to ${\mathcal{Z}}_{d,G}$ and perhaps to some other similar sets.

Note that if ${\mathcal{A}},{\mathcal{B}}$ contain dense ${\mathcal{G}}_{\delta}$ sets, then their intersection ${\mathcal{A}}\cap{\mathcal{B}}$ also contains a dense ${\mathcal{G}}_{\delta}$ set.

We remark that the proof of [10, Theorem 1.3] implies that for any integer $d\geqslant 2$ the set $\Xi_{d}$ contains a dense ${\mathcal{G}}_{\delta}$ set, where

[TABLE]

Meanwhile, the proof of [10, Theorem 1.6] implies that the similar statement also holds for the monomial sums $G_{d}(x;N)$ with any integer $d\geqslant 2$. Therefore for integer $d\geqslant 2$, we conclude that there are dense ${\mathcal{G}}_{\delta}$ sets of $\mathbf{x}\in\mathsf{T}_{d}$ and $x\in T$ with

[TABLE]

and for any $\varepsilon>0$

[TABLE]

respectively.

It is natural to ask about the Lebesgue measure and the Hausdorff dimensions of the sets ${\mathcal{Z}}_{d,W}$ and ${\mathcal{Z}}_{d,G}$.

Definition 1.11** (Hausdorff dimension).**

The Hausdorff dimension of a set ${\mathcal{A}}\subseteq{\mathbb{R}}^{d}$ is defined as

[TABLE]

For the properties of the Hausdorff dimension and its applications we refer to [16, 36].

We now give lower bounds on the Hausdorff dimensions of ${\mathcal{Z}}_{d,W}$ and ${\mathcal{Z}}_{d,G}$.

Theorem 1.12**.**

For any integer $d\geqslant 2$, we have

[TABLE]

The above lower bounds on the Hausdorff dimension are perhaps far from the truth but we do not know how to improve them.

1.4. Outline of the method

We first show that the quadratic Weyl sums and monomial sums take small values at some rational points and thus in their neighbourhood. Secondly, we apply some results and tools from metric number theory to show that the neighbourhoods of these rational points are large sets in the sense of topology and Hausdorff dimension.

We conclude this paper with a brief outline of some further directions of research and some additional ideas which may lead to improvements of our results. In particular, we formulate several conjectures on the behaviour of Weyl sums.

2. Exponential sums

2.1. Notation and conventions

As usual, the notations $U=O(V)$, $U\ll V$ and $V\gg U$ are equivalent to $|U|\leqslant c|V|$ for some positive constant $c$. Throughout the paper, all implied constants are absolute.

The letter $p$ always denotes a prime number.

2.2. Complete and incomplete Gaussian sums

For a prime $p$, let ${\mathbb{F}}_{p}$ denote the finite field of $p$ elements, which we identify with the set $\{0,\ldots,p-1\}$. Furthermore, let

[TABLE]

We also need the following bound for the incomplete Gaussian sums, see [29].

Lemma 2.1**.**

For each prime $p$ and integer $b\neq 0$ we have

[TABLE]

From Lemma 2.1 we immediately derive the following.

Lemma 2.2**.**

For any prime $p$ and any $a,b\in{\mathbb{F}}_{p}$ with $b\neq 0$ we have

[TABLE]

We emphasise that the implied constant in Lemma 2.2 is absolute.

We now recall that the Gaussian sum modulo $2p$ or $4p$ exhibit very different behaviour and sometimes vanish.

Lemma 2.3**.**

Let $p\geqslant 3$ and $a,b\in{\mathbb{Z}}$.

• (i)

If $\gcd(b,2p)=1$ then

[TABLE]

• (ii)

If $\gcd(ab,2p)=1$ then

[TABLE]

Proof.

Part (i) is well known and follows instantly from the explicit formula for Gauss sums, see [25, Theorem 3.4].

For Part (ii), we note

[TABLE]

since for an odd $a$ we have ${\mathbf{\,e}}_{4p}\left(2ap\right)=\exp(\pi ia)=-1$. The result now follows.     \sqcap$$\sqcup

2.3. Monomial sums

Here we always assume that $d\geqslant 3$. We have the following analogues of Lemma 2.3, which follows from the trivial observation that the map $x\mapsto x^{d}$ is a permutation of ${\mathbb{F}}_{p}$ provided that $\gcd(d,p-1)=1$.

Lemma 2.4**.**

Let an integer $d\geqslant 3$ and a prime $p\geqslant 3$ be such that $\gcd(d,p-1)=1$. Then for any integer $b$ with $\gcd(b,p)=1$ one has

[TABLE]

Using the Weil bound together with the standard completion technique, see [25, Sections 11.11 and 12.2] we also immediately obtain:

Lemma 2.5**.**

For any prime $p$ and any $a\in{\mathbb{F}}_{p}\setminus\{0\}$ we have

[TABLE]

We again emphasise that the implied constant in Lemma 2.5 is absolute.

2.4. Continuity of Weyl sums

We present our next result in a much more general form than we need for purpose, but we believe in this generality it may have other applications.

Lemma 2.6**.**

Let integer $N\geqslant 1$ and a vector $\mathbf{x}\in\mathsf{T}_{d}$ be such that for any $M\leqslant N$ we have

[TABLE]

for some real non-negative $\alpha$, $\kappa$ and $K$. Then for any positive $\tau=O(1)$ and $\mathbf{y}\in\mathsf{T}_{d}$ with

[TABLE]

we have

[TABLE]

where the implied constant is absolute.

Proof.

Let $\delta_{i}=y_{i}-x_{i}$, $i=1,\ldots,d$. For each $n\in{\mathbb{N}}$ we have

[TABLE]

It follows that

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For each $k\in{\mathbb{N}}$ we now turn to the estimate

[TABLE]

where $\xi_{n}=\delta_{1}n+\ldots+\delta_{d}n^{d}$. Applying partial sum formula we derive

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where

[TABLE]

By our assumption, we obtain

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and also, observing that the sequence $\xi_{M}$ is monotonically non-decreasing, we have

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We now derive

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We see from (2.2) and (2.3) that

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which together with (2.1) yields the desired bound.     \sqcap$$\sqcup

For the convenience of our applications we formulate several specialisations of Lemma 2.6 with different choices of $(\alpha,\kappa,K)$.

For example, with $(\alpha,\kappa,K)=(0,0,K)$ we obtain the following result on approximations to rational sums.

Corollary 2.7**.**

Let integer $N\geqslant 1$ and integer $m\geqslant 2$. Let integer $d\geqslant 3$ and

[TABLE]

be such that for any $n\leqslant N$ we have

[TABLE]

for some real non-negative $K$. Then for any $\tau>0$ and $\mathbf{x}\in\mathsf{T}_{d}$ with

[TABLE]

we have

[TABLE]

where the implied constant is absolute.

If $d=2$ and $\mathbf{x}=(a/p,b/p)\in\mathsf{T}_{2}$ for some integers $a$ and $b$ with $\gcd(b,p)=1$, then by Lemma 2.2 we can take $(\alpha,\kappa,K)=(1,p^{-1/2},p^{1/2})$ in Lemma 2.6.

3. Proofs of results on small sums

3.1. Proof of Theorem 1.10

We first show the sets ${\mathcal{Z}}_{2,W},{\mathcal{Z}}_{2,G}$ and ${\mathcal{Z}}_{d,G}$ (with $d\geqslant 3$) are dense ${\mathcal{G}}_{\delta}$ sets. We introduce some notation. For $p\geqslant 3$ let

[TABLE]

Furthermore, let

[TABLE]

and

[TABLE]

For ${\mathcal{A}}\subseteq{\mathbb{R}}^{d}$ the $\delta$-neighbourhood of ${\mathcal{A}}$ is defined as

[TABLE]

Clearly for any $p\geqslant 3$ the functions $S_{2}(\mathbf{x};4p)$, $G(x;2p)$ and $G_{d}(x;p)$ are continuous function with respect to the variables $\mathbf{x}\in\mathsf{T}_{2}$ and $x\in\mathsf{T}$. Thus, applying Lemmas 2.3 and 2.4, we derive that for any $p$ and $\eta>0$ there exists $\delta_{p,\eta}>0$ such that

• •

for any $\mathbf{x}\in{\mathcal{P}}_{p}(\delta_{p,\eta})$ we have $|S_{2}(\mathbf{x};4p)|\leqslant\eta$;

• •

for $d=2$ and any $x\in\ {\mathcal{Q}}_{p}(\delta_{p,\eta})$ we have $|G(x;2p)|\leqslant\eta$;

• •

for $d\geqslant 3$, $\gcd(d,p-1)=1$ and any $x\in{\mathcal{R}}_{p}(\delta_{p,\eta})$ we have $|G_{d}(x;p)|\leqslant\eta$.

Using these notation and (3.1), we set

[TABLE]

and also for $d\geqslant 3$

[TABLE]

Observe that

[TABLE]

and for any $k$ the set

[TABLE]

are dense open sets, which finishes the proof for the sets ${\mathcal{Z}}_{2,W},{\mathcal{Z}}_{2,G}$ and ${\mathcal{Z}}_{d,G}$ (with $d\geqslant 3$).

We now turn to the claim that the set ${\mathcal{Z}}_{d,W}$ with $d\geqslant 3$ is a dense ${\mathcal{G}}_{\delta}$ set in $\mathsf{T}_{d}$. This is essentially contained in [10, Remark 2.8], for the completeness we present the complete argument to here.

For $d\geqslant 3$ and a prime number $p$ with $\gcd(d,p-1)=1$, the map: $x\to x^{d}$ permutes ${\mathbb{F}}_{p}$. Hence, for any $\lambda\in{\mathbb{F}}_{p}\setminus\{0\}$ we have

[TABLE]

It follows that for any $\lambda\in{\mathbb{F}}_{p}\setminus\{0\}$ we have

[TABLE]

We call the set ${\mathfrak{U}}\subseteq{\mathbb{F}}_{p}^{d}$ a discrete box with the side length $\ell({\mathfrak{U}})=L$ if

[TABLE]

where the set ${\mathcal{I}}_{j}=\{k_{j}+1,\ldots,k_{j}+L\}$ is a set of $L\leqslant p$ consecutive integers, (reduced modulo $p$ if $k_{j}+L\geqslant p$) for each $j=1,\ldots,d$.

Assuming $p>d$ we see that

[TABLE]

By [10, Lemma 2.5] for any discrete box ${\mathfrak{B}}\subseteq{\mathbb{F}}_{p}^{d}$ with side length $\ell({\mathfrak{B}})\geqslant Cp^{1-1/2d}\log p$ for some constant $C$ there exists $\lambda\in{\mathbb{F}}_{p}\setminus\{0\}$ such that

[TABLE]

Combining with (3.2) we deduce that for any box ${\mathcal{B}}\subseteq\mathsf{T}_{d}$ with side length larger than $5Cp^{-1/2d}\log p$ there exists a point $\mathbf{x}\in{\mathcal{B}}\cap{\mathcal{Z}}_{d,W}$. Combining with the well known fact that there are infinitely many primes $p$ such that $(d,p-1)=1$ (for instance this follows by applying Dirichlet’s theorem on arithmetic progressions), we conclude that for any $k\in{\mathbb{N}}$ the set

[TABLE]

is a dense subset of $\mathsf{T}_{d}$. By using the continuity of the function $S_{d}(\mathbf{x};N)$ and applying the similar arguments as for the sets ${\mathcal{Z}}_{2,W},{\mathcal{Z}}_{2,G}$ and ${\mathcal{Z}}_{d,G}$ (with $d\geqslant 3$), we obtain the result.

3.2. Proof of Theorem 1.12

We first note that the lower bound for $\dim_{H}{\mathcal{Z}}_{d,W}$ with $d\geqslant 3$ follows from the fact that

[TABLE]

and the monotonicity of the Hausdorff dimension

[TABLE]

Furthermore directly from the definition of the Hausdorff dimension, we see that

[TABLE]

Thus in the following we only show the lower bounds on the Hausdorff dimension of the sets ${\mathcal{Z}}_{2,W}$ and ${\mathcal{Z}}_{d,G}$. Our method is a modification of the argument as in the proof of Theorem 1.10 for the sets ${\mathcal{Z}}_{2,W},{\mathcal{Z}}_{2,G}$ and ${\mathcal{Z}}_{d,G}$ (with $d\geqslant 3$).

In analogy of Lemma 2.2 we have the following. Using the completion techniques, see [25, Section 12.2], similarly to Lemma 2.5 we immediately obtain that or each prime $p\geqslant 3$ and $a,b\in{\mathbb{F}}_{p}\setminus\{0\}$ with $\gcd(ab,2p)=1$ we have

[TABLE]

We remark that perhaps one can also remove $\log p$ from the bound (3.3) and have a full analogue of Lemma 2.2, but this does not affect our final result.

Let ${\mathcal{P}}^{*}$ be the collection of point $(x_{1},x_{2})\in\mathsf{T}_{2}$ that there are infinitely many $p$ and $a,b$ such that $\gcd(ab,2p)=1$ and

[TABLE]

For these $(x_{1},x_{2})$ by applying (3.3) and Corollary 2.7 with $N=4p,K=\sqrt{p}\log p$ and $\tau=\left(p^{1/2}(\log p)^{2}\right)^{-1}$, we deduce that

[TABLE]

By Lemma 2.3 (ii) we have

[TABLE]

and hence

[TABLE]

For Gaussian sums let $a\in{\mathbb{F}}_{p}\setminus\{0\}$ and $\gcd(a,2p)=1$. Then by [29] we have

[TABLE]

Let ${\mathcal{Q}}^{*}$ be the collection of point $x\in[0,1)$ such that there are infinitely many $p$ and $a\in{\mathbb{F}}_{p}\setminus\{0\}$ with $\gcd(a,2p)=1$ and

[TABLE]

For this $x$ and $a/p$ by (3.6) and Corollary 2.7 with $N=2p,K=\sqrt{p}$ and $\tau=\left(p^{1/2}\log p\right)^{-1}$, we obtain

[TABLE]

By Lemma 2.3 (i) we have

[TABLE]

and hence

[TABLE]

Now we turn to the monomial sums for $d\geqslant 3$. Let ${\mathcal{R}}^{*}$ be the collection of point $x\in[0,1)$ such that there are infinitely many $p$ and $a\in{\mathbb{F}}_{p}\setminus\{0\}$ with $\gcd(d,p-1)=1$ and

[TABLE]

For this $x$ and $a/p$, using Lemma 2.5 and Corollary 2.7 with $N=p$, $K=\sqrt{p}\log p$ and $\tau=(p^{1/2}(\log p)^{2})^{-1}$, we derive that

[TABLE]

Combining with Lemma 2.4 we have

[TABLE]

and hence

[TABLE]

On the other hand, it follows essentially from the Jarník–Besicovitch theorem [2, Section 6] we obtain

[TABLE]

and also for $d\geqslant 3$

[TABLE]

Combining with (3.5), (3.7) and  (3.8) and using the monotonicity of the Hausdorff dimension we derive the result. We omit the details here, but refer to [16, Theorem 10.3] and [10] for the closely related arguments.

We remark that we need the prime number theorem for the lower bounds of $\dim_{H}{\mathcal{P}}^{*},\dim_{H}{\mathcal{Q}}^{*}$, see [16, Theorem 10.3] for details (and also [10]). Similarly, for the lower bounds of $\dim_{H}{\mathcal{R}}^{*}$ we need the prime number theorem for arithmetic progressions in a very week form that for all large enough $x$ one has

[TABLE]

4. Further results, open problems and conjectures

4.1. An approach to improve the lower bound for $\dim_{H}{\mathcal{Z}}_{d,W}$

We use the notation from the proof of Theorem 1.12. For $d=2$ the “box condition” (3.4) can be extended to the below “rectangle condition” given by (4.1) below. To be precise, let $\mathfrak{R}$ be the collection of point $(x_{1},x_{2})\in\mathsf{T}_{2}$ that there are infinitely many $p$ and $a,b$ such that $\gcd(ab,2p)=1$ and

[TABLE]

Clearly we have

[TABLE]

Moreover for $(x_{1},x_{2})\in\mathfrak{R}$ by applying (3.3) and Corollary 2.7 with

[TABLE]

we deduce that

[TABLE]

By Lemma 2.3 (ii) we have

[TABLE]

and hence

[TABLE]

Thus we have a larger subset of ${\mathcal{Z}}_{2,W}$, however we do not know how to deal with the Hausdorff dimension of $\mathfrak{R}$.

Furthermore for $\dim_{H}{\mathcal{Z}}_{d,W}$ with $d\geqslant 3$ we conjecture that one could improve the Hausdorff dimension of the set ${\mathcal{Z}}_{d,W}$ by taking a direct way instead of the arguments at beginning of the proof of Theorem 1.12.

We remark that the results and techniques of [6, 7] can shed some light on improving Lemma 2.6 and similar “continuity properties” of Weyl sums, and thus may lead improvement to improvements of the bound of Theorem 1.12 on the dimension of ${\mathcal{Z}}_{d,W}$.

4.2. The topology of Weyl sums

For $\mathbf{x}\in\mathsf{T}_{d}$ we define the orbit of $\mathbf{x}$ as

[TABLE]

We remark that our interest to orbits is partially motivated by classical works of Lehmer [32], Loxton [33, 34] and Forrest [20, 21], as well as by more recent results of Cellarosi and Marklof [8], Fayad [17], Greschonig, Nerurkar and Volný [22]. In the discrete settings, that is, for rational exponential sum, similar questions have been considered by Demirci Akarsu [12, 13], Demirci Akarsu and Marklof [14], Kowalski and Sawin [30, 31], Ricotta and Royer [38], Ricotta, Royer and the second author [39].

From (1.2) and (1.3) we obtain that there is a dense ${\mathcal{G}}_{\delta}$ set of $\mathbf{x}\in\mathsf{T}_{d}$ such that the zero of $\mathbb{C}$ is a accumulation point of $O_{d}(\mathbf{x})$, and the set $O_{d}(\mathbf{x})$ is unbounded (alternatively, we can say that the infinity is an accumulation point of the set $O_{d}(\mathbf{x})$).

Modelling the sums $S_{d}(\mathbf{x};N)$ by the sums of $N$ independent and uniform distributed random complex vectors from the unit circle, it is natural to make the following:

Conjecture 4.1**.**

For almost all $\mathbf{x}\in\mathsf{T}_{d}$ in the sense of Lebesgue measure the orbit $O_{d}(\mathbf{x})$ is everywhere dense in $\mathbb{C}$.

We remark that for some point $\mathbf{x}\in\mathsf{T}_{d}$ the set $O_{d}(\mathbf{x})$ is not dense in $\mathbb{C}$. For instance, Theorem 1.1 (ii) implies that if the continued fraction of $x_{2}$ has bounded partial quotients then for any $x_{1}\in\mathsf{T}$ the set $O_{2}(\mathbf{x})$ is not a dense subset of $\mathbb{C}$.

We show two more examples in the following that the set $O_{2}(\mathbf{x})$ have a biased distribution. From Lemma 2.3 (ii) and (3.3) for any integer $1\leqslant a,b\leqslant p-1$ with $\gcd(ab,2p)=1$ and any $N\in{\mathbb{N}}$ we obtain

[TABLE]

It follows that the set $O_{2}(a/4p,b/4p)$ is bounded.

Now we turn to another example. From Lemma 2.2 for any $1\leqslant a,b\leqslant p-1$ and any $N>p$ we have

[TABLE]

Since $S_{d}((a/p,b/p);p)$ is a constant, we deduce that the infinity of $\mathbb{C}$ is the only accumulation point of the set $O_{2}(a/p,b/p)$. Moreover the set $O_{2}(a/p,b/p)$ is contained in some tube of $\mathbb{C}$ with width nearly $\sqrt{p}$, that is, recalling the definition of ${\mathcal{A}}(\delta)$ at (3.1),

[TABLE]

where ${\mathcal{L}}$ is some line of $\mathbb{C}$ and $C$ is some absolute positive constant.

4.3. Restricted Weyl sums

For an integer $d\geqslant 2$ and a real $\alpha\in(0,1)$ we denote

[TABLE]

From (1.1) we deduce that for any $\alpha\in(1/2,1)$ one has $\lambda_{d}({\mathcal{E}}_{d,\alpha})=0$, where $\lambda_{d}$ is the $d$-dimensional Lebesgue measure.

Note that Theorem 1.3 implies that $\lambda_{2}({\mathcal{E}}_{2,1/2})=1$.

Motivated from the works on Diophantine approximation on manifold, see [1, 3, 24] and references therein, we consider the following very general question. We remark that the below set ${\mathcal{X}}$ can be some fractal set.

Question 4.2**.**

For a given set ${\mathcal{X}}\subseteq\mathsf{T}_{d}$ equipped with some measure $\mu$ what can we say about

[TABLE]

provided ${\mathcal{X}}$ has some natural geometric, algebraic or combinatorial structure?

For the following special case when ${\mathcal{X}}\subset\mathsf{T}_{2}$ is a parabola, we make:

Conjecture 4.3** (Dimension).**

Let ${\mathcal{X}}=\{(t,t^{2}):\leavevmode\nobreak\ t\in[0,1]\}$ then

[TABLE]

Conjecture 4.4** (Measure).**

Let ${\mathcal{X}}=\{(t,t^{2}):\leavevmode\nobreak\ t\in[0,1]\}$ and let $\mu$ be the natural probability measure on ${\mathcal{X}}$. Then one has $\mu({\mathcal{E}}_{2,1/2}\cap{\mathcal{X}})=1$.

It is also natural to ask similar questions about the intersection ${\mathcal{E}}_{d,\alpha}$ with the moment curve, that is,

[TABLE]

4.4. Random Cantor sets

Now we turn to the case where ${\mathcal{X}}$ is some random fractal set. We consider $d=2$ and a simpler model of random Cantor sets to show our ideas. We start by informal description of the model, see [16, Chapter 15] for more related constructions (see also [9] for the detailed construction and reference therein).

We remark that many other random fractals are also called random Cantor sets.

We apply the following iterative procedure:

• •

We divide the unit square $[0,1]^{2}$ into four equal interior disjoint closed squares in a natural way such that each of these four squares has side length $1/2$, We choose uniformly at random remove one square, and let ${\mathcal{E}}_{1}^{\omega}$ be the collection of the three remaining squares.

• •

For each of the remaining squares, we apply the same procedure and obtain a collection ${\mathcal{E}}_{2}^{\omega}$ of nine squares.

• •

We continue inductively in the same manner by dividing each square into four squares and then uniformly and independently at randomly remove one, getting a collection ${\mathcal{E}}_{n}^{\omega}$ of $3^{n}$ squares.

Clearly, each square in ${\mathcal{E}}_{n}^{\omega}$ has the side length $2^{-n}$.

Definition 4.5** (Random Cantor set).**

A random Cantor set is

[TABLE]

Let $\Omega$ be our probability space which consists of all the possible outcomes of these random limit sets.

Now for each random Cantor set ${\mathcal{E}}^{\omega}\in\Omega$ we associate a natural measure on ${\mathcal{E}}^{\omega}$. The desired measure should give each squares of ${\mathcal{E}}_{n}^{\omega}$ the same mass, which is $1/3^{n}$. To be precise, let ${\mathcal{E}}^{\omega}=\bigcap_{n=1}^{\infty}{\mathcal{E}}_{n}^{\omega}$ be a realization. For each $n\in{\mathbb{N}}$ define the measure

[TABLE]

where $\textbf{1}_{{\mathcal{F}}}$ is the indicator function of the set ${\mathcal{F}}$. Note that for every square ${\mathcal{Q}}$ of ${\mathcal{E}}_{n}^{\omega}$ we have

[TABLE]

Note that the sequence of the measure $\mu_{n}^{\omega},n\in{\mathbb{N}}$ weakly convergence to a measure $\mu^{\omega}$, see [36, Chapter 1]. We call this measure $\mu^{\omega}$ the natural measure on ${\mathcal{E}}^{\omega}$.

For our application, we need the following Lemma 4.6, suggested by Pablo Shmerkin (private communication). For completeness we present the complete proof here. In the following we use $\lambda_{d}$ to denote the $d$-dimensional Lebesgue measure.

Lemma 4.6**.**

Let ${\mathcal{F}}\subseteq[0,1]^{d}$ with $\lambda_{d}({\mathcal{F}})=0$ then almost surely (for ${\mathcal{E}}^{\omega}\in\Omega$) we have $\mu^{\omega}({\mathcal{F}})=0$.

Proof.

We use $\mathbb{E}$ and $\mathbb{P}$ as the notation of expectation and probability, respectively.

Let $\varepsilon>0$, then there is an open set ${\mathcal{U}}\supset{\mathcal{F}}$ with $\lambda_{d}({\mathcal{U}})<\varepsilon$. Applying Fubini’s theorem, see [36, Theorem 1.14], we obtain

[TABLE]

The last identity holds by using the fact that for all $\mathbf{x}\in{\mathcal{U}}$ (with an exceptional set with zero Lebesgue measure, except some grid line) one has

[TABLE]

Moreover by [36, Theorem 1.24] we have

[TABLE]

Putting all together and applying Fatou’s lemma [15, Theorem 1.17], we derive

[TABLE]

By the arbitrary choice of $\varepsilon>0$ we finish the proof.     \sqcap$$\sqcup

Combining Lemma 4.6 with Theorem 1.3 we obtain

Corollary 4.7**.**

Almost surely for ${\mathcal{E}}^{\omega}\in\Omega$ and for $\mu^{\omega}$-almost all $\mathbf{x}\in{\mathcal{E}}^{\omega}$ we have

[TABLE]

We remark that Lemma 4.6 indeed holds for a variety families of random fractals, and hence the Corollary 4.7 also follows immediately.

Acknowledgement

We would like to thank Jens Marklof for informing us on some bounds of quadratic Weyl sums and also application of a dynamical system approach to the distribution of quadratic Weyl sums. We are grateful to Dzmitry Badziahin and Mumtaz Hussain for helpful discussions and some additional references. We also thank Pablo Shmerkin for the idea of Lemma 4.6.

This work was supported by ARC Grant DP170100786.