New bounds of Weyl sums

TL;DR

This paper improves bounds on Weyl sums and discrepancy of fractional parts of polynomials by extending Wooley's method with new ideas, including a self-improving approach and generalizations to projections, also providing Hausdorff dimension bounds.

Contribution

It introduces novel bounds for Weyl sums, extends results to projections of coefficients, and develops a self-improving method for tighter bounds.

Findings

Enhanced metric bounds on Weyl sums

New bounds on discrepancy of polynomial fractional parts

Upper bounds on Hausdorff dimension of large Weyl sum sets

Abstract

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \ldots , u_d)$ onto a lower dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a ``self-improving'' approach, which leads an infinite series of monotonically decreasing bound, converging to our final result.

Equations

$$S_d(\mathbf{u};N)=\sum _{n=1}^N\mathbf{e}(u_{1}n+\dots +u_d{n}^d),\mathbf{u}=(u_1,\dots ,u_d)\in {\mathsf{T}}_d,$$

$$\mathbf{e}(x)=\exp (2\pi ix).$$

$$u_{1}n+\dots +u_d{n}^d,n\in \mathbb{N},$$

$$\left|u_{\nu }-\frac{a}{q}\right|⩽\frac{1}{q^2}.$$

$$|S_d(\mathbf{u};N)|⩽C(\epsilon )N^{1+\epsilon }{\left(q^{-1}+N^{-1}+q{N}^{-\nu }\right)}^{1/d(d-1)}.$$

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

$$s(q)=\frac{q(q+1)}{2}.$$

$$T_{\mathbf{a},\mathbf{\phi }}(\mathbf{u};N)=\sum _{n=1}^N{a}_n\mathbf{e}\left(u_1{\phi }_1(n)+\dots +u_d{\phi }_d(n)\right).$$

$${\mathsf{T}}_d={\mathsf{T}}_k\times {\mathsf{T}}_{d-k}$$

$$T_{\mathbf{a},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)=\sum _{n=1}^N{a}_n\mathbf{e}\left(\sum _{j=1}^k{x}_j{\phi }_j(n)+\sum _{j=1}^{d-k}{y}_j{\phi }_{k+j}(n)\right).$$

$$T_{\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)=T_{\mathbf{e},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N).$$

$$\{{\phi }_1(T),\dots ,{\phi }_d(T)\}=\{T,\dots ,T^d\}$$

$$\underset{\mathbf{y}\in {\mathsf{T}}_{d-k}}{\sup }|T_{\mathbf{\phi }}(\mathbf{u},\mathbf{y};N)|⩽N^{{\Gamma }_{\ast }(\mathbf{\phi },k)+o(1)},N\to \infty ,$$

$${\Gamma }_{\ast }(\mathbf{\phi },k)=\frac{1}{2}+\frac{2{\sigma }_k(\mathbf{\phi })+d-k+1}{2d^2+4d-2k+2}$$

$${\sigma }_k(\mathbf{\phi })=\sum _{j=k+1}^d\mathrm{deg}{\phi }_j.$$

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

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

$$W(T;\mathbf{\phi })=\det {\left({\phi }_i^{(j-1)}(T)\right)}_{i,j=1}^n$$

$$\underset{\mathbf{y}\in {\mathsf{T}}_{d-k}}{\sup }|T_{\mathbf{a},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)|⩽N^{\Gamma (\mathbf{\phi },k)+o(1)},N\to \infty ,$$

$$\Gamma (\mathbf{\phi },k)=\frac{1}{2}+\frac{2{\sigma }_k(\mathbf{\phi })+d-k}{2d^2+4d-2k}.$$

$$\Gamma (\mathbf{\phi },k)<{\Gamma }_{\ast }(\mathbf{\phi },k),k=1,\dots ,d.$$

$$\left|\sum _{n=1}^N{a}_n\mathbf{e}(u_1{\phi }_1(n)+\dots +u_d{\phi }_d(n))\right|⩽N^{1/2+o(1)},N\to \infty .$$

$$\underset{\mathbf{y}\in {\mathsf{T}}_{d-k}}{\sup }|T_{\mathbf{a},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)|⩽\underset{\hat{\mathbf{y}}\in {\mathsf{T}}_{d-k+1}}{\sup }|T_{\mathbf{a},\mathbf{\phi }}(\check{\mathbf{x}},\hat{\mathbf{y}};N)|.$$

$$\underset{\mathbf{y}\in {\mathsf{T}}_{d-k}}{\sup }|T_{\mathbf{a},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)|⩽\underset{\hat{\mathbf{y}}\in {\mathsf{T}}_{d-k+1}}{\sup }|T_{\mathbf{a},\hat{\mathbf{\phi }}}(\mathbf{x},\hat{\mathbf{y}};N)|.$$

$$\underset{k<j⩽d}{\min }\mathrm{deg}{\phi }_j=1.$$

$$\underset{\mathbf{y}\in {\mathsf{T}}_{d-k}}{\sup }|T_{\mathbf{a},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)|⩽N^{{\Gamma }_{YL}(\mathbf{\phi },k)+o(1)},N\to \infty ,$$

$${\Gamma }_{YL}(\mathbf{\phi },k)=\frac{1}{2}+\frac{{\sigma }_k(\mathbf{\phi })}{2s(d)}.$$

$${\Gamma }_{YL}(\mathbf{\phi },k)<\Gamma (\mathbf{\phi },k).$$

$$\underset{1⩽j⩽k}{\min }\mathrm{deg}{\phi }_j=1.$$

$$\underset{\mathbf{y}\in {\mathsf{T}}_{d-k}}{\sup }|T_{\mathbf{a},\mathbf{\phi }}(\mathbf{x},\mathbf{y};N)|⩽N^{{\Gamma }_{XL}(\mathbf{\phi },k)+o(1)},N\to \infty ,$$

Full text

We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_{1},\ldots,u_{d})$ onto a lower dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a “self-improving” approach, which leads to an infinite series of monotonically decreasing bounds, converging to our final result.

1. 1 Introduction

2. 1.1 Background

3. 1.2 Set-up and previous results

4. 1.3 An overview of our results and tools

5. 2 Main results

6. 2.1 Results for coordinate-wise projections of $\mathbf{u}$: a traditional point of view

7. 2.2 Results for arbitrary orthogonal projections of $\mathbf{u}$: a new point of view

8. 2.3 Uniform distribution modulo one

9. 3 Preliminaries

10. 3.1 Notation and conventions

11. 3.2 Generalised mean value theorems

12. 3.3 The completion technique

13. 3.4 Continuity of exponential sums

14. 3.5 Distribution of large values of exponential sums

15. 3.6 Orthogonal projections of boxes

16. 3.7 Orthogonal projections and large values of exponential sums

17. 4 Proofs of exponential sum bounds

18. 4.1 Proof of Theorem 2.1

19. 4.2 Proof of Theorem 2.3

20. 4.3 Proof of Theorem 2.6

21. 4.4 Proof of Theorem 2.9

22. 5 Proof of discrepancy bounds

23. 5.1 Preliminaries

24. 5.2 Proof of Theorem 2.14

25. 5.3 Proof of Theorem 2.16

26. 6 Comments

27. 6.1 Discrepancy of polynomials

28. 6.2 The structure of the exceptional sets

29. 6.3 Further possible extensions

1. Introduction

1.1. Background

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

[TABLE]

have been introduced and estimated by Weyl [20], and thus are called the Weyl sums, where throughout the paper we denote

[TABLE]

By investigating the properties of the sums (1.1), Weyl [20] established the uniformity of distribution modulo one of the sequence

[TABLE]

provided at least one of the coefficients $u_{1},\ldots,u_{d}$ is irrational. The Weyl sums play crucial role in many other fundamental number theoretic problems. These include estimating the zero-free region of the Riemann zeta-function and thus obtaining good bounds in the error term in the prime number theorem, see [16, Section 8.5], and the Waring problem, see [16, Section 20.2] or [19] for a more detailed treatment. Further problems include bounds of very short character sums modulo highly composite numbers [16, Section 12.6] and various problems from the uniformity of distribution theory and Diophantine approximations [2].

However, despite more than a century long history of estimating such sums, the behaviour of individual sums is not well understood. There have been several conjectures made about their behaviour and true order of magnitude of such sums depending on Diophantine properties of the coefficients $u_{1},\ldots,u_{d}$; some have been ruled out, some are still widely open even in the case of sums with monomials $un^{d}$, see [6, 7].

The following bound is a direct implication of the current form of the Vinogradov mean value theorem from [5, 21] and is explicitly given in [4, Theorem 5]. Let $\mathbf{u}=(u_{1},\ldots,u_{d})\in\mathsf{T}_{d}$ be such that for some $\nu$ with $2\leqslant\nu\leqslant d$ and some positive integers $a$ and $q$ with $\gcd(a,q)=1$ we have

[TABLE]

Then for any $\varepsilon>0$ there exits a constant $C(\varepsilon)$ such that

[TABLE]

It seems that the current bounds are expected to be far away from the true size of $S_{d}(\mathbf{u};N)$. We also remark that as mentioned by Bourgain [4, Section 3], for $d\leqslant 6$ better results are known.

On the other hand, the behaviour of the average value of the Weyl sums has recently been fully unveiled in works of Bourgain, Demeter and Guth [5] (for $d\geqslant 4$) and Wooley [21] (for $d=3$) (see also [23]) in the best possible form

[TABLE]

of the Vinogradov mean value theorem, where for $q\in{\mathbb{R}}$ we denote

[TABLE]

Here we study a question which originates from the work of Flaminio and Forni [14] and has also been studied in more detail by Wooley [22]. Namely, here we seek results which hold for all values of the components of $\mathbf{u}=(u_{1},\ldots,u_{d})\in\mathsf{T}_{d}$ on some prescribed set of positions and almost all values of the components on the remaining positions. Thus this question “interpolates” between individual bounds and bounds involving some kind of averaging. Wooley [22, Theorem 1.1] has shown that in this setting the individual bound in (1.2) can be improved. In this project we introduce several additional arguments and make further improvements.

1.2. Set-up and previous results

Given a family $\bm{\varphi}=\left(\varphi_{1},\ldots,\varphi_{d}\right)\in{\mathbb{Z}}[T]^{d}$ of $d$ distinct nonconstant polynomials and a sequence of complex weights $\mathbf{a}=(a_{n})_{n=1}^{\infty}$, for $\mathbf{u}=(u_{1},\ldots,u_{d})\in\mathsf{T}_{d}$ we define the trigonometric polynomials

[TABLE]

Furthermore, for $k=1,\ldots,d$, decomposing

[TABLE]

with $\mathsf{T}_{k}=[0,1)^{k}$ and $\mathsf{T}_{d-k}=[0,1)^{d-k}$. Given $\mathbf{x}\in\mathsf{T}_{k}$, $\mathbf{y}\in\mathsf{T}_{d-k}$ we refine the notation (1.5) and write

[TABLE]

If $\mathbf{a}=\mathbf{e}=(1)_{n=1}^{\infty}$ (that is, $a_{n}=1$ for each $n\in{\mathbb{N}}$) we just write

[TABLE]

For the classical case $a_{n}=1$ for all $n\in{\mathbb{N}}$ and the polynomials

[TABLE]

satisfying some natural necessary conditions, the result of Wooley [22, Theorem 1.1] together with the modern knowledge towards the Vinogradov mean value theorem, see (1.3), asserts that for almost all $\mathbf{x}\in\mathsf{T}_{k}$ with respect to the $k$-dimensional Lebesgue measure on $\mathsf{T}_{k}$, one has

[TABLE]

where

[TABLE]

and

[TABLE]

We remark that the bound (1.7) is presented in a more explicit form than in [22, Theorem 1.1] as we have used the optimal result of Wooley [23, Theorem 1.1] for the parameter $u$ of [22, Theorem 1.1]. Furthermore the results in [22, Theorem 1.1] have the restriction that $k<d$, but our method works for $k=d$ also. Naturally, for the case $k=d$ we consider $\mathbf{x}=\mathbf{u}$ only and remove the variable $\mathbf{y}$ from each statement for this special case.

Here we use some new ideas to extend the method and results of Wooley [22] in serval directions. In particular, we obtain an improvement of (1.7).

We note that it is also interesting to find a tight upper bound for the almost all points $\mathbf{u}\in\mathsf{T}_{d}$ for the classical Weyl sums $S_{d}(\mathbf{u};N)$ given by (1.1). In this direction the authors [8, Appendix A] have shown that for almost all $\mathbf{u}\in\mathsf{T}_{d}$ one has

[TABLE]

It is very natural to conjecture that the exponent $1/2$ cannot be improved.

Fedotov and Klopp [13, Theorem 0.1] have shown that the conjecture is true for $d=2$. More precisely, for any non-decreasing sequence $\{g(n)\}_{n=1}^{\infty}$ of positive numbers, for almost all $\mathbf{u}\in\mathbb{T}_{2}$ we have

[TABLE]

We remark that the conjecture is still open for $d\geqslant 3$.

As in [22], we give applications of our bounds of exponential sums to bounds on the discrepancy (see Section 2.6 for a definition) of the sequence of fractional parts of polynomials. However, we modify and improve the approach of Wooley [22] of passing from exponential sums to the discrepancy and obtain stronger results.

1.3. An overview of our results and tools

Here we obtain results of three different types:

• (i)

We study the scenario of Wooley [22] when the vector $\mathbf{u}\in\mathsf{T}_{d}$ is split into two parts $\mathbf{x}$ and $\mathbf{y}$ formed by its components which is related to the coordinate-wise projections of $\mathbf{u}\in\mathsf{T}_{d}$.

• (ii)

We introduce and study an apparently new problem related to arbitrary orthogonal projections of $\mathbf{u}\in\mathsf{T}_{d}$. As an additional benefit, our results for arbitrary orthogonal projections, combined with the classical Marstrand–Mattila projection theorem, see [17, Chapter 5], lead to an upper bound on the Hausdorff dimension of sets or large Weyl sums, complementing our previous lower bounds [8].

• (iii)

As in [22], we study the uniform distribution of polynomials modulo one and obtain a bound for the discrepancy, which improves that of [22, Theorem 1.4].

We note that although our results improve those of [22], we see the main value of this work in new ways to combine several principal elements which have been used in the area for quite some time. Namely, we exploit the interplay between

• (i)

the modern form of the Vinogradov mean value theorem, see, for example, Lemma 3.1;

• (ii)

the completion technique, see, for example, Lemma 3.2;

• (iii)

continuity of Weyl sums, see, for example, Lemmas 3.4, 3.5 and 3.6, which in turn leads us to a new type of “self-improving” results in Lemma 4.1 and Corollary 4.2.

As we have mentioned, as one of the applications of our results we obtain an upper bound for the Hausdorff dimension of sets with large Weyl sums, see Corollaries 2.11 and 2.12. Other applications are given in Theorems 2.6 and 2.16 to a bound on short Weyl sums and to the distribution of fractional parts of polynomials over short intervals, respectively.

We hope that similar combinations of these ideas may find several other applications. We also believe that the idea of studying arbitrary orthogonal projections and its applications to bounds of Hausdorff dimension has never been used in analytic number theory before this work.

2. Main results

2.1. Results for coordinate-wise projections of $\mathbf{u}$: a traditional point of view

Throughout the paper, let $\bm{\varphi}=\left(\varphi_{1},\ldots,\varphi_{d}\right)\in{\mathbb{Z}}[T]^{d}$ be $d$ distinct nonconstant polynomials such that the Wronskian

[TABLE]

does not vanish identically and let $\mathbf{a}=(a_{n})_{n=1}^{\infty}$ be a sequence of complex weights with $a_{n}=n^{o(1)}$.

We start with a very broad generalisation of (1.7). We recall that $\sigma_{k}(\bm{\varphi})$ is given by (1.8).

Theorem 2.1**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically, then for almost all $\mathbf{x}\in\mathsf{T}_{k}$ one has

[TABLE]

where

[TABLE]

We remark that Theorem 2.1 gives a nontrivial upper bound provided that $\sigma_{k}(\bm{\varphi})<s(d)$, where $s(d)$ is given by (1.4).

Furthermore for the classical choice of $\bm{\varphi}$ as in (1.6) we always have $\sigma_{k}(\bm{\varphi})<s(d)$. Elementary calculations show that

[TABLE]

Thus Theorem 2.1 gives a direct improvement and generalisation of the bound (1.7), which is due to Wooley [22, Theorem 1.1].

We observe also that $\Gamma(\bm{\varphi},k)=1/2$ when $k=d$, and this gives the same bound as that of (1.9) for more general polynomials $\bm{\varphi}$. More precisely, we have the following.

Corollary 2.2**.**

Let $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$, $d\geqslant 2$, be such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically and let $a_{n}=n^{o(1)}$, then for almost all $\mathbf{u}\in\mathsf{T}_{d}$ one has

[TABLE]

For some special cases of $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$, we obtain a series of better bounds which in almost all cases are better than Theorem 2.1 and thus give a further improvement of the result of Wooley [22, Theorem 1.1]. The bounds are based on a new “self-improving” argument, see Lemma 4.1 and Corollary 4.2 below.

We consider the following three mutually exclusive possibilities:

• A.

For some $k+1\leqslant j\leqslant d$ we have $\deg\varphi_{j}=1$, that is, with there is a linear polynomial attached to the vector $\mathbf{y}$.

• B.

For some $1\leqslant j\leqslant k$ we have $\deg\varphi_{j}=1$, that is, with there is a linear polynomial attached to the vector $\mathbf{x}$.

• C.

For all $1\leqslant j\leqslant d$ we have $\deg\varphi_{j}\geqslant 2$, that is, $\bm{\varphi}$ does not contain a linear polynomial.

To reflect these there possibilities we denote new exponents, replacing $\Gamma(\bm{\varphi},k)$ by $\Gamma_{YL}(\bm{\varphi},k)$, $\Gamma_{XL}(\bm{\varphi},k)$ and $\Gamma_{NL}(\bm{\varphi},k)$.

In fact our main result below Theorem 2.3 handles only Case A. Then we reduce Cases B and C to Case A.

Indeed, for Case B, assuming without loss of generality that $\deg\varphi_{k}=1$, we simply write $T_{\mathbf{a},\bm{\varphi}}(\mathbf{x},\mathbf{y};N)=T_{\mathbf{a},\bm{\varphi}}(\check{\mathbf{x}},\hat{\mathbf{y}};N)$, where we append $x_{k}$ to $\mathbf{y}$ so that $\hat{\mathbf{y}}=(x_{k},y_{k+1},\ldots,y_{d})\in\mathsf{T}_{d-k+1}$ which we estimate for almost all $\check{\mathbf{x}}=(x_{1},\dots,x_{k-1})\in\mathsf{T}_{k-1}$. That is, in Case B, for any $\mathbf{x}\in\mathsf{T}_{k}$ we use the inequality

[TABLE]

To tackle Case C, we simply replace $T_{\mathbf{a},\bm{\varphi}}(\mathbf{x},\mathbf{y};N)$ with $T_{\mathbf{a},\widehat{\bm{\varphi}}}(\mathbf{x},\widehat{\mathbf{y}};N)$, where we append $y_{d+1}$ to $\mathbf{y}$ and $\varphi_{d+1}(T)=T$ to $\bm{\varphi}$, so that $\widehat{\mathbf{y}}=(y_{k+1},\ldots,y_{d},y_{d+1})\in\mathsf{T}_{d-k+1}$ and $\widehat{\bm{\varphi}}=(\varphi_{1},\ldots,\varphi_{d},\varphi_{d+1})\in{\mathbb{Z}}[T]^{d+1}$, which we estimate for almost all $\mathbf{x}=(x_{1},\dots,x_{k})\in\mathsf{T}_{k}$. That is, in Case C, for any $\mathbf{x}\in\mathsf{T}_{k}$ we use the inequality

[TABLE]

More precisely, recalling the definitions (1.4) and (1.8) in Case A we have the following bound.

Theorem 2.3**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically and suppose that

[TABLE]

Then for almost all $\mathbf{x}\in\mathsf{T}_{k}$ one has

[TABLE]

where

[TABLE]

We remark that if $\sigma_{k}(\bm{\varphi})<d(d+1)/2$ then for each $k=1,\ldots,d-1$ one has

[TABLE]

Moreover for the case $k=d$ we have $\Gamma_{YL}(\bm{\varphi},d)=\Gamma(\bm{\varphi},d)=1/2$. Thus Theorem 2.3 improves Theorem 2.1 if there is a linear polynomial attached to the vector $\mathbf{y}$.

As we have described in the above, for Cases B and C, from Theorem 2.3 we obtain the following two estimates:

Corollary 2.4**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically and suppose that $k\geqslant 2$ and

[TABLE]

Then for almost all $\mathbf{x}\in\mathsf{T}_{k}$ one has

[TABLE]

where

[TABLE]

Corollary 2.5**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\widehat{\bm{\varphi}})$ of $\widehat{\bm{\varphi}}=(\varphi_{1},\ldots,\varphi_{d},\varphi_{d+1})\in{\mathbb{Z}}[T]^{d+1}$ with $\varphi_{d+1}(T)=T$ does not vanish identically and suppose that

[TABLE]

Then for almost all $\mathbf{x}\in\mathsf{T}_{k}$ one has

[TABLE]

where

[TABLE]

As yet another application of Theorem 2.3 we derive the following bounds for the short sums. For $M\in{\mathbb{Z}}$, we consider Weyl sums over short intervals

[TABLE]

Theorem 2.6**.**

For almost all $x_{d}\in[0,1]$, one has

[TABLE]

*where $\mathbf{u}=(y_{1},\ldots,y_{d-1},x_{d})$. *

From Theorem 2.6 we immediately derive that for almost all $\mathbf{u}\in\mathsf{T}_{d}$ one has

[TABLE]

Note that using the bound (1.2) and a similar observation about the leading coefficient of shifted polynomials, one obtains a version of Theorem 2.6 with the exponent

[TABLE]

Remark 2.7**.**

The bounds of Theorem 2.3 and Corollaries 2.4 and 2.5 are typically stronger than that of Theorem 2.1. However in the case when $\min_{j=1,\ldots,d}\deg\varphi_{j}\geqslant 2$ but $W(T;\widehat{\bm{\varphi}})=0$ this is the only result at our disposal.

2.2. Results for arbitrary orthogonal projections of $\mathbf{u}$: a new point of view

We now consider other projections which seems to be a new scenario which has not been studied in the literature prior to this work.

We need to introduce some notation first.

Let ${\mathcal{G}}(d,k)$ denote the collections of all the $k$-dimensional linear subspaces of ${\mathbb{R}}^{d}$. For ${\mathcal{V}}\in{\mathcal{G}}(d,k)$, let $\pi_{\mathcal{V}}:\leavevmode\nobreak\ {\mathbb{R}}^{d}\to{\mathcal{V}}$ denote the orthogonal projection onto ${\mathcal{V}}$. For $0<\alpha<1$, we consider the set

[TABLE]

We also use $\lambda\left({\mathcal{S}}\right)$ to denote the Lebesgue measure of ${\mathcal{S}}\subseteq\mathsf{T}_{d}$ (and also for sets in other spaces).

We are interested in the following apparently new point of view:

Question 2.8**.**

Given $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$, for what $\alpha$ we have $\lambda(\pi_{\mathcal{V}}({\mathcal{E}}_{\mathbf{a},\bm{\varphi},\alpha}))=0$ for all ${\mathcal{V}}\in{\mathcal{G}}(d,k)$?

We now see that Theorem 2.1 implies that for $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is as in Theorem 2.1 and $a_{n}=n^{o(1)}$, for any

[TABLE]

we have

[TABLE]

where $\pi_{d,k}$ is the orthogonal projection of $\mathsf{T}_{d}$ onto $\mathsf{T}_{k}$, that is,

[TABLE]

For the degree sequence $\deg\varphi_{1},\ldots,\deg\varphi_{d}$ we denote them as

[TABLE]

and define

[TABLE]

We remark that the following result is similar to the result of Theorem 2.1, with the change of $\widetilde{\sigma}_{k}(\bm{\varphi})$ only.

Theorem 2.9**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically and $\widetilde{\sigma}_{k}(\bm{\varphi})<d(d+1)/2$, then for any ${\mathcal{V}}\in{\mathcal{G}}(d,k)$ one has

[TABLE]

provided that $\alpha>\widetilde{\Gamma}(\bm{\varphi},k)$ where

[TABLE]

We now consider Question 2.8 in the classical case (1.6) and the sums (1.1). That is, we study the following set

[TABLE]

which we define for $0<\alpha<1$ and integer $d\geqslant 2$. Note that in this setting we have $\widetilde{\sigma}_{k}(\bm{\varphi})=(d+k+1)(d-k)/2$.

Corollary 2.10**.**

For any ${\mathcal{V}}\in{\mathcal{G}}(d,k)$ one has

[TABLE]

provided that $\alpha>\widetilde{\Gamma}_{d,k}$ where

[TABLE]

We remark that the orthogonal projection of sets is a fundamental topic in fractal geometry and geometric measure theory. Recall the classical Marstrand–Mattila projection theorem: Let ${\mathcal{A}}\subseteq{\mathbb{R}}^{d}$, $d\geq 2,$ be a Borel set with Hausdorff dimension $s$, see [17, Chapter 5] for more details and related definitions. Then we have:

• •

Dimension part: If $s\leq k$, then the orthogonal projection of ${\mathcal{A}}$ onto almost all $k$-dimensional subspaces has Hausdorff dimension $s$.

• •

Measure part: If $s>k$, then the orthogonal projection of ${\mathcal{A}}$ onto almost all $k$-dimensional subspaces has positive $k$-dimensional Lebesgue measure.

From the Marstrand–Mattila projection theorem and Corollary 2.10 we obtain the following results. For ${\mathcal{A}}\subseteq{\mathbb{R}}^{d}$ we use $\dim{\mathcal{A}}$ to denote the Hausdorff dimension of ${\mathcal{A}}$.

Corollary 2.11**.**

Let $k,d$ be two integers with $1\leqslant k<d$ and $d\geqslant 2$. Then $\dim{\mathcal{E}}_{d,\alpha}\leqslant k$ for any

[TABLE]

In particular, taking $k=d-1$ we obtain

Corollary 2.12**.**

For any integer $d\geqslant 2$ one has $\dim{\mathcal{E}}_{d,\alpha}\leqslant d-1$ for any

[TABLE]

We note that the authors [9] showed that for any $\alpha\in(1/2,1)$ one has

[TABLE]

with some explicit function $u(d,\alpha)<d$. Moreover the function

[TABLE]

However the exact comparison between the bound $u(d,\alpha)$ and that of Corollary 2.11 is not immediately obvious.

We remark that the authors [8] have obtained a lower bound of the Hausdorff dimension of ${\mathcal{E}}_{d,\alpha}$. Among other things, it is shown in [8] that for any $d\geqslant 2$ and $\alpha\in(0,1)$ one has

[TABLE]

with some explicit function $\xi(d,\alpha)>0$. As a counterpart to (2.5), we remark that we expect $\dim{\mathcal{E}}_{d,\alpha}=d$ for $\alpha\in(0,1/2)$, see also [8, 10]. On the other hand, we do not have any plausible conjecture about the exact behaviour of $\dim{\mathcal{E}}_{d,\alpha}$ for $\alpha\in[1/2,1)$.

Remark 2.13**.**

In principle, one can obtain various analogues of Theorem 2.3 and Corollaries 2.4 and 2.5 for the arbitrary projections. However they require imposing some additional (and rather cluttered) restrictions on linear combinations of components of $\bm{\varphi}$. We omit these similar but more involuted arguments for this setting.

2.3. Uniform distribution modulo one

Let $\xi_{n}$, $n\in{\mathbb{N}}$, be a sequence in $[0,1)$. The discrepancy of this sequence at length $N$ is defined as

[TABLE]

We note that sometimes in the literature the scaled quantity $N^{-1}D_{N}$ is called the discrepancy, but since our argument looks cleaner with the definition (2.6), we adopt it here.

For $\mathbf{x}\in\mathsf{T}_{k}$, $\mathbf{y}\in\mathsf{T}_{d-k}$ we consider the sequence

[TABLE]

and for each $N$ we denote by $D_{\bm{\varphi}}(\mathbf{x},\mathbf{y};N)$ the corresponding discrepancy of its fractional parts.

Wooley [22, Theorem 1.4] has proved that ($d\geqslant 3$) for almost all $\mathbf{x}\in\mathsf{T}_{k}$ with $1\leqslant k\leqslant d-1$ one has

[TABLE]

where

[TABLE]

We improve this bound as follows.

Theorem 2.14**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically and $\sigma_{k}(\bm{\varphi})<d(d+1)/2$, then for almost all $\mathbf{x}\in\mathsf{T}_{k}$ one has

[TABLE]

where

[TABLE]

For the classical choice of $\bm{\varphi}$ as in (1.6) we always have $\sigma_{k}(\bm{\varphi})<s(d)$, where $s(d)$ is given by (1.4), and elementary calculations show that

[TABLE]

Thus, as before with Theorem 2.1, we see that Theorem 2.14 gives a direct improvement and generalisation of the result of Wooley [22, Theorem 1.4].

Remark 2.15**.**

It is natural to try to obtain analogues of the bounds of exponential sums of Theorem 2.3 and Corollaries 2.4 and 2.5 for the discrepancy. However our main tool, the Erdős–Turán inequality, see Lemma 5.1 below, involves a growing with $N$ family of exponential sums of length $N$. So one needs some additional ideas to adjust our argument to this case.

From Theorem 2.14 we derive a bound on the discrepancy of real polynomials over short intervals. More precisely, we now given an upper bound on $D_{d}(\mathbf{u};M,N)$ which denotes the discrepancy of the sequence of fractional parts

[TABLE]

Theorem 2.16**.**

For almost all $x_{d}\in[0,1]$, one has

[TABLE]

where $\mathbf{u}=(y_{1},\ldots,y_{d-1},x_{d})$.

From Theorem 2.16 we obtain that for almost all $\mathbf{u}\in\mathsf{T}_{d}$ one has

[TABLE]

Finally, for $D_{d}(\mathbf{u};N)$ we claim that by combining [15, Theorem 5.13] with some additional arguments, one can show that for almost all $\mathbf{u}\in\mathsf{T}_{d}$ with $d\geqslant 2$ one has the following stronger bound,

[TABLE]

We give a proof at Section 6. Furthermore we conjecture that this upper bound is the best possible except for a logarithm factor. We remark that this is true for $d=2$ which follows by applying a result of Fedotov and Klopp [13, Theorem 0.1] and the Koksma inequality [15, Theorem 5.4]. However the conjecture is still open when $d\geqslant 3$.

3. Preliminaries

3.1. Notation and conventions

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

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 $V$ is large enough. One additional advantage of using $V^{o(1)}$ is that it absorbs $\log V$ and other similar quantities without changing the whole expression.

We use $\#{\mathcal{S}}$ to denote the cardinality of a finite set ${\mathcal{S}}$.

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

We say that some property holds for almost all $\mathbf{x}\in[0,1)^{k}$ if it holds for a set ${\mathcal{X}}\subseteq[0,1)^{k}$ of $k$-dimensional Lebesgue measure $\lambda({\mathcal{X}})=1$.

We always assume that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ consists of polynomials $\varphi_{j}$ of degrees

[TABLE]

3.2. Generalised mean value theorems

For the classical case of the Weyl sums $S_{d}(\mathbf{u};N)$ as in (1.1), the Parseval identity gives

[TABLE]

Furthermore, we have the Vinogradov mean value theorem, in the optimal form (1.3).

We use the following a general form due to Wooley [23, Theorem 1.1], which extends the bound (1.3) to the sums $T_{\mathbf{a},\bm{\varphi}}(\mathbf{u};N)$.

We also recall that for functions $\bm{\psi}=\left(\psi_{1},\ldots,\psi_{d}\right)\in{\mathbb{Z}}[T]^{d}$ their Wronskian $W(T;\bm{\psi})$ is defined in (2.1).

Lemma 3.1**.**

For any a family $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ of $d$ polynomials such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically, any sequence of complex weights $\mathbf{a}=(a_{n})_{n=1}^{\infty}$, and any integer $N\geqslant 1$, we have the upper bound

[TABLE]

for any real positive $\sigma\leqslant s(d)$, where $s(d)$ is given by (1.4).

3.3. The completion technique

We remark that the completion technique has many applications in analytic number theory. We show the following version for the later application.

Lemma 3.2**.**

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

[TABLE]

where

[TABLE]

Proof.

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

[TABLE]

Observe that by the orthogonality

[TABLE]

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

[TABLE]

see [16, Equation (8.6)]. It follows that

[TABLE]

which finishes the proof.     \sqcap$$\sqcup

For $\mathbf{x}\in[0,1)^{k}$, $\mathbf{y}\in[0,1)^{d-k}$, by Lemma 3.2 we also have

[TABLE]

where

[TABLE]

Note that for any $N$ there exists a sequence $b_{N}(n)$ such that

[TABLE]

and $W_{\mathbf{a},\bm{\varphi}}(\mathbf{u};N)$ can be written as

[TABLE]

Indeed, since each inner sums in $W_{\mathbf{a},\bm{\varphi}}(\mathbf{u};N)$ depends only on $h$ (for a fixed $N$) we clearly can write

[TABLE]

for some complex $\vartheta_{h}$ on the unit circle. Hence we can take

[TABLE]

in (3.2). Combining (3.2) with Lemma 3.1 we obtain the following.

Corollary 3.3**.**

Let $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically and $a_{n}=n^{o(1)}$, then we have

[TABLE]

3.4. Continuity of exponential sums

We start with the following general statement which could be of independent interest.

Lemma 3.4**.**

Let integer $N\geqslant 1$ and a vector $\mathbf{u}=(u_{1},\ldots,u_{d})\in\mathsf{T}_{d}$ be such that for any $1\leqslant M\leqslant N$ we have

[TABLE]

as $N\to\infty$, for some real $\rho\geqslant 0$. Then for any positive $\omega=O(1)$ and $\mathbf{v}=(v_{1},\ldots,v_{d})\in\mathsf{T}_{d}$ with

[TABLE]

if $\varphi_{i}(n)>0$ for all large enough n, and

[TABLE]

if $\varphi_{i}(n)<0$ for all large enough $n$, we obtain

[TABLE]

where the implied constant is absolute.

Proof.

We first remark that the condition $\varphi_{i}(n)>0$ for all sufficiently large $n$ is equivalent to that the polynomial $\varphi_{i}(n)$ is eventually an increasing function, which is used below when we apply the partial sum formula.

Furthermore we remark that the choice of $\mathbf{v}=(v_{1},\ldots,v_{d})$ is to guarantee the “non-negativity condition”

[TABLE]

for all large enough $n$.

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

[TABLE]

It follows that

[TABLE]

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

[TABLE]

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

[TABLE]

where

[TABLE]

By our assumption, we obtain

[TABLE]

Observe that there exists a constant $M_{0}$ (which depends on $\bm{\varphi}$ only) such that the sequence $\xi_{M}$ is monotonically non-decreasing for all $M\geqslant M_{0}$. It follows that

[TABLE]

We see from (3.5) and (3.6) that

[TABLE]

which together with (3.4) yields

[TABLE]

Since $|\exp(\omega)-1|\ll\omega$ for $\omega=O(1)$, the desired result follows.     \sqcap$$\sqcup

We remark that if $a_{n}=n^{o(1)}$ we can always apply Lemma 3.4 with $\rho=1$, which we actually do in Lemma 3.5 below. On the other hand, we can use some $0<\rho<1$ for some special cases, see Lemma 3.6 below. Furthermore, for applications of Lemma 3.4 to Lemmas 3.5 and 3.6, the value $\omega$ is quite small, in particular, $\omega=o(1)$.

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 box centred at $\mathbf{u}$ and with the side lengths $2\bm{\zeta}$ by

[TABLE]

We have the following analogues of Wooley [22, Lemma 2.1].

Lemma 3.5**.**

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ and $e_{1},\ldots e_{d}$ are as (3.1). Let $0<\alpha<1$ and let $\varepsilon>0$ be sufficiently small. Suppose that $|W_{\mathbf{a},\bm{\varphi}}(\mathbf{u};N)|\geqslant N^{\alpha}$ for some $\mathbf{u}\in\mathsf{T}_{d}$, then for

[TABLE]

there is a set ${\mathcal{R}}^{*}(\mathbf{u},\bm{\zeta})\subseteq{\mathcal{R}}(\mathbf{u},\bm{\zeta})$ with

[TABLE]

such that

[TABLE]

holds for any $\mathbf{v}\in{\mathcal{R}}^{*}(\mathbf{u},\bm{\zeta})$ provided that $N$ is large enough.

Proof.

Let ${\mathcal{R}}^{*}(\mathbf{u},\bm{\zeta})$ be the set of vectors $\mathbf{v}\in{\mathcal{R}}(\mathbf{u},\bm{\zeta})$ which satsify the “non-negativity condition” (3.3). By Lemma 3.4, applied with $\rho=1$ and $\omega=N^{\alpha-1-\varepsilon}$, for $\mathbf{v}\in{\mathcal{R}}^{*}(\mathbf{u},\bm{\zeta})$ we have

[TABLE]

The result follows from the definition of $W_{\mathbf{a},\bm{\varphi}}(\mathbf{v};N)$ in Lemma 3.2.     \sqcap$$\sqcup

Lemma 3.6**.**

Let $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ be such that

[TABLE]

Let $0<\alpha<t\leqslant 1$ and let $\varepsilon>0$ be sufficiently small. Suppose that $|W_{\mathbf{a},\bm{\varphi}}(\mathbf{u};N)|\geqslant N^{\alpha}$ for some $\mathbf{u}=(\mathbf{x},\mathbf{y})\in\mathsf{T}_{d}$ and

[TABLE]

for some constant $C$, then for

[TABLE]

there is a set ${\mathcal{R}}^{*}(\mathbf{u},\bm{\zeta})\subseteq{\mathcal{R}}(\mathbf{u},\bm{\zeta})$ with

[TABLE]

such that

[TABLE]

*holds for any $\mathbf{v}\in{\mathcal{R}}^{*}(\mathbf{u},\bm{\zeta})$ provided that $N$ is large enough. *

Proof.

From (3.7), without loss of generality, we assume that $\deg\varphi_{d}=1$ and hence

[TABLE]

for some real numbers $\varrho_{1},\varrho_{2}$ with $\varrho_{1}\neq 0$. For any integer $h$ we write

[TABLE]

For any $1\leqslant M\leqslant N$, by (3.8), with the vector of coefficients

[TABLE]

we obtain

[TABLE]

Combining this with (3.9) we derive

[TABLE]

By Lemma 3.4, applied with the coefficients $a_{n}{\mathbf{\,e}}(hn/N)$ instead of $a_{n}$, $\rho=t$ and $\omega=N^{\alpha-t-\varepsilon}$, we have

[TABLE]

By the definition of $W_{\mathbf{a},\bm{\varphi}}(\mathbf{v};N)$ in Lemma 3.2 we obtain

[TABLE]

the result now follows for all large enough $N$.     \sqcap$$\sqcup

We note that a similar concept of continuity of Weyl sums has also played a major role in a different point of view on the distribution of Weyl sums [6, 7].

3.5. Distribution of large values of exponential sums

We adapt the arguments of [22, Lemma 2.2] to our setting.

First we show the following useful box counting result. We note that any better bound of the exponent of $N$ immediately yields an improvement of our results.

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

[TABLE]

where $e_{1},\ldots e_{d}$ are as (3.1).

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

[TABLE]

boxes of the form

[TABLE]

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

[TABLE]

Lemma 3.7**.**

In the above notation, we have

[TABLE]

Proof.

Let ${\mathcal{R}}\in\mathfrak{R}$. By Lemma 3.5 if $|W_{\mathbf{a},\bm{\varphi}}(\mathbf{u};N)|\geqslant N^{\alpha}$ for some $\mathbf{u}\in{\mathcal{R}}$, then there is a set ${\mathcal{R}}^{*}\subseteq{\mathcal{R}}$ with

[TABLE]

such that for any $\mathbf{v}\in{\mathcal{R}}^{*}$ we have $|W_{\mathbf{a},\bm{\varphi}}(\mathbf{v};N)|\geqslant N^{\alpha}/2$. Combining this with Corollary 3.3 we have

[TABLE]

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

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

Corollary 3.8**.**

Let $0<\alpha<1$. Then

[TABLE]

Proof.

We fix some sufficiently small $\varepsilon>0$ and define the set

[TABLE]

Observe that

[TABLE]

Clearly we have

[TABLE]

By Lemma 3.7 and the choice of $\bm{\zeta}$ in (3.10), since $\varepsilon$ is arbitrary, we now obtain the desired result.     \sqcap$$\sqcup

Applying Lemma 3.6, in analogy with Corollary 3.8, we obtain the following.

Corollary 3.9**.**

Let $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ be such that

[TABLE]

Let $\Omega\subseteq\mathsf{T}_{k}$ with $\lambda(\Omega)>0$ and let $0<\alpha<t\leqslant 1$. Suppose that there exits a positive constant $C$ such that for all $\mathbf{x}\in\Omega$ we have

[TABLE]

Let

[TABLE]

then we obtain

[TABLE]

Proof.

We fix some sufficiently small $\varepsilon>0$ and define

[TABLE]

and we divide $\mathsf{T}_{d}$ into $U^{*}$ rectangles in a natural way. Let $\mathfrak{R}^{*}$ be the collection of these rectangles.

We also define an analogue of $\widetilde{\mathfrak{R}}$ (3.11) as

[TABLE]

Using Lemma 3.6 instead of Lemma 3.5 in the proof Lemma 3.7 we obtain

[TABLE]

Applying the similar argument as in Corollary 3.8 we obtain the result.     \sqcap$$\sqcup

3.6. Orthogonal projections of boxes

We start with the following general result which is perhaps well known.

Lemma 3.10**.**

Let ${\mathcal{R}}\subseteq{\mathbb{R}}^{d}$ be a box with the side lengths $h_{1}\geqslant\ldots\geqslant h_{d}$. Then for all ${\mathcal{V}}\in{\mathcal{G}}(d,k)$ we have

[TABLE]

where the implied constant depends on $d$ and $k$ only.

Proof.

The idea is to cover a box by balls, and use that the size of the orthogonal projections of any given ball does not depend on the  choice of ${\mathcal{V}}\in{\mathcal{G}}(d,k)$.

More precisely, without loss of generality we can assume that

[TABLE]

Let

[TABLE]

be a subset of ${\mathcal{R}}$. Since for any $\mathbf{x}\in{\mathcal{R}}$ there exists $\mathbf{y}\in{\mathcal{R}}_{k}$ such that

[TABLE]

we obtain

[TABLE]

where ${\mathcal{B}}({\bf 0},dh_{k+1})$ is the ball of ${\mathbb{R}}^{d}$ centred at ${\bf 0}$ and of radius $dh_{k+1}$ and for ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathbb{R}}^{d}$, as usual, we define:

[TABLE]

Now we intend to cover ${\mathcal{R}}_{k}$ by a family of balls of ${\mathbb{R}}^{d}$ such that each of these balls has the radius roughly $h_{k+1}$.

For each $1\leqslant j\leqslant k$ we have

[TABLE]

where ${\mathcal{I}}_{j,q}=[qh_{k+1},(q+1)h_{k+1})$ and

[TABLE]

Then

[TABLE]

Observe that for each choice on integers $q_{1},\ldots,q_{k}$ with

[TABLE]

there exists a ball ${\mathcal{B}}_{q_{1},\ldots,q_{k}}$ of ${\mathbb{R}}^{d}$ of radius $dh_{k+1}$ such that

[TABLE]

Denote the collection of these balls by

[TABLE]

It follows that

[TABLE]

Since the radius of each ball ${\mathcal{B}}\in{\mathfrak{B}}$ is $dh_{k+1}$, we have

[TABLE]

where $2{\mathcal{B}}(\mathbf{x},r)={\mathcal{B}}(\mathbf{x},2r)$. Together with (3.13) and (3.15) we obtain

[TABLE]

It follows that for any ${\mathcal{V}}\in{\mathcal{G}}(d,k)$ we have

[TABLE]

Since for each ball ${\mathcal{B}}\in{\mathfrak{B}}$ the projection $\pi_{{\mathcal{V}}}(2{\mathcal{B}})$ is a ball of the $k$-dimensional subspace ${\mathcal{V}}$ with radius $2dr_{k+1}$, one has

[TABLE]

Combining this with (3.14), we obtain

[TABLE]

which gives the result.     \sqcap$$\sqcup

3.7. Orthogonal projections and large values of exponential sums

We now provide a basic tool for the proof of Theorem 2.9. Applying Lemma 3.7 and Lemma 3.10 we obtain the following analogue of Corollary 3.8.

Corollary 3.11**.**

Let $0<\alpha<1$. For any ${\mathcal{V}}\in{\mathcal{G}}(d,k)$ we have

[TABLE]

as $N\to\infty$, where $\widetilde{\sigma}_{k}(\bm{\varphi})$ is given by (2.4).

Proof.

We fix some sufficiently small $\varepsilon>0$. We use the same notation as in Section 3.5, including the choice of $\zeta_{j}$, $j=1,\ldots,d$ in (3.10). For ${\mathcal{R}}\in\mathfrak{R}$ with the side lengths $\zeta_{1},\ldots,\zeta_{d}$ we denote them as

[TABLE]

For $j=1,\ldots,d$ by (2.3) we obtain

[TABLE]

We also define the set

[TABLE]

Observe that

[TABLE]

Combining this with Lemma 3.7, Lemma 3.10 and (3.16) we obtain

[TABLE]

By the definition of $\widetilde{\sigma}_{k}(\bm{\varphi})$ and since $\varepsilon$ is arbitrary, we obtain the desired bound.     \sqcap$$\sqcup

4. Proofs of exponential sum bounds

4.1. Proof of Theorem 2.1

We fix some $\alpha>1/2$ and set

[TABLE]

We now consider the set

[TABLE]

By Corollary 3.8 we have

[TABLE]

We ask that the parameters satisfy the following condition

[TABLE]

Combining (4.2) with the Borel–Cantelli lemma, we obtain that

[TABLE]

Since

[TABLE]

we conclude that for almost all $\mathbf{x}\in\mathsf{T}_{k}$ there exists $i_{\mathbf{x}}$ such that for any $i\geqslant i_{\mathbf{x}}$ one has

[TABLE]

We fix this $\mathbf{x}$ in the following arguments. For any $N\geqslant N_{i_{\mathbf{x}}}$ there exists $i$ such that

[TABLE]

By Lemma 3.2 and (4.3) we have

[TABLE]

Note that the condition (4.2) can be written as

[TABLE]

which gives the desired bound.

4.2. Proof of Theorem 2.3

Recall that

[TABLE]

Applying a similar chain of arguments as the proof of Theorem 2.1, from Corollary 3.9 we derive the following “self-improving” property of Weyl sums.

Lemma 4.1**.**

Let $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ be such that

[TABLE]

Let $\Omega\subseteq\mathsf{T}_{k}$ with $\lambda(\Omega)>0$ and let $0<t\leqslant 1$. Suppose that there exits a positive constant $C$ such that for all $\mathbf{x}\in\Omega$ we have

[TABLE]

Then for almost all $\mathbf{x}\in\Omega$ and for any $\varepsilon>0$ there exists a positive constant $C(\mathbf{x},\varepsilon)$ such that

[TABLE]

where

[TABLE]

Proof.

We fix some $0<\alpha<t$ and set

[TABLE]

For each $i$ denote

[TABLE]

Corollary 3.9 gives

[TABLE]

Similarly to the proof of Theorem 2.1, we ask the parameters satisfy the condition

[TABLE]

which is

[TABLE]

Thus we finishes the proof.     \sqcap$$\sqcup

We remark that in Lemma 4.1 if $t>\Gamma_{YL}(\bm{\varphi},k)$ then $f(t)<t$, this is reason why we call it a “self-improving” type result.

We now immediately derive from Lemma 4.1 the following “self-improving” property underlying our bounds. Compared to Lemma 4.1 it allows us to have some level of non-uniformity in our assumption.

Corollary 4.2**.**

Let $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ be such that

[TABLE]

Let $\Omega\subseteq\mathsf{T}_{k}$ with $\lambda(\Omega)>0$ and let $0<t\leqslant 1$. Suppose that for almost all $\mathbf{x}\in\Omega$ there exits a positive constant $C(\mathbf{x})$ such that

[TABLE]

Then for almost all $\mathbf{x}\in\Omega$ and for any $\varepsilon>0$ there exists a positive constant $C(\mathbf{x},\varepsilon)$ such that

[TABLE]

where

[TABLE]

Proof.

We take a decomposition $\Omega=\bigcup_{q=0}^{\infty}\Omega_{q}$ such that $\lambda(\Omega_{0})=0$ and for each $q\geqslant 1$ the sums are uniformly bounded by $qN^{t}$, that is, for any $\mathbf{x}\in\Omega_{q}$ we have

[TABLE]

Applying Lemma 4.1 for each $\Omega_{q}$, $q\geqslant 1$, we obtain the desired result.     \sqcap$$\sqcup

Now we turn to the proof of Theorem 2.3. Denote

[TABLE]

Firstly Theorem 2.1 claims that for almost all $\mathbf{x}\in\mathsf{T}_{k}$ and for any $\varepsilon>0$ there exists a constant $C(\mathbf{x},\varepsilon)$ such that

[TABLE]

Applying Corollary 4.2 repeatedly, we obtain the following sequence

[TABLE]

Since the function $t\mapsto f(t)$ is strictly monotonically decreasing for

[TABLE]

and

[TABLE]

by an arbitrary choice small enough $\varepsilon>0$ at each steps, we finish the proof.

4.3. Proof of Theorem 2.6

For $M\in{\mathbb{Z}}$, recall that Weyl sums over short intervals are defined as follows

[TABLE]

We write

[TABLE]

and observe that in the polynomial identity

[TABLE]

where for $j=0,1,\ldots,d-1$, each $v_{j}$, depends only on $u_{1},\ldots,u_{d}$ and $M$.

Hence Theorem 2.3, applied $k=1$, $\varphi_{1}(T)=T^{d}$ and thus with $\sigma_{1}(\bm{\varphi})=d(d-1)/2$, yields the desired estimate on $S_{d}(\mathbf{u};M,N)$.

4.4. Proof of Theorem 2.9

As we have claimed, Theorem 2.9 now follows by applying Corollary 3.11 instead of Corollary 3.8 and using similar arguments as in the proofs of Theorem 2.1. We omit these very similar arguments here.

5. Proof of discrepancy bounds

5.1. Preliminaries

We start with recalling the classical Erdős–Turán inequality (see, for instance, [11, Theorem 1.21]).

Lemma 5.1**.**

Let $\xi_{n}$, $n\in{\mathbb{N}}$, be a sequence in $[0,1)$. Then for the discrepancy $D_{N}$ given by (2.6) and any $G\in{\mathbb{N}}$, we have

[TABLE]

We also use the following trivial property of the Lebesgue measure, see [22, Section 3] for a short proof.

Lemma 5.2**.**

Let ${\mathcal{A}}\subseteq\mathsf{T}_{d}$ and $g\in{\mathbb{N}}$, then

[TABLE]

5.2. Proof of Theorem 2.14

As in Section 3.5, if $\mathbf{a}=\mathbf{e}=(1)_{n=1}^{\infty}$, we just write

[TABLE]

Let $N_{i}=2^{i}$, $i\in{\mathbb{N}}$ and let $G_{i}=\left\lfloor N_{i}^{\eta}\right\rfloor$ for some $\eta>0$ to be chosen later. For each $g=1,\ldots,G_{i}$ let

[TABLE]

and

[TABLE]

Observe that

[TABLE]

where the notation ${\mathcal{B}}_{i}$ is given by (3.12) in the case $\mathbf{a}=\mathbf{e}$. By Lemma 5.2 and the inequality (4.1) we conclude that

[TABLE]

We ask that the fixed $\alpha$ and $\eta$ satisfy the following condition

[TABLE]

Combining this with the Borel–Cantelli lemma, and choosing a small enough $\varepsilon$, we obtain that

[TABLE]

It follows that for almost all $\mathbf{x}\in\mathsf{T}_{k}$ there exists $i_{\mathbf{x}}$ such that for any $i\geqslant i_{\mathbf{x}}$ and any $g=1,\ldots,G_{i}$, one has

[TABLE]

Combining with Lemma 3.2 we obtain that for any $N\geqslant N_{i_{\mathbf{x}}}$ there exists $i\in{\mathbb{N}}$ such that

[TABLE]

and for any $g=1,\ldots,G_{i}$, one has

[TABLE]

Applying Lemma 5.1 for $N$ and $G=G_{i}$ we conclude that

[TABLE]

Let $\eta=1-\alpha$. The condition (5.1) can be written as

[TABLE]

which finishes the proof.

5.3. Proof of Theorem 2.16

Recall that $D_{d}(\mathbf{u};M,N)$ is the discrepancy of the sequence of fractional parts

[TABLE]

Clearly this sequence is same as

[TABLE]

and thus as before, see (4.4), we see that this sequence is the same as

[TABLE]

where for $j=0,1,\ldots,d-1$, each $v_{j}$, depends only on $u_{1},\ldots,u_{d}$ and $M$. Furthermore let $\mathbf{u}^{*}=(v_{1},\ldots,v_{d-1},u_{d})$ and

[TABLE]

Then we have

[TABLE]

where the implied constant is absolute. This can be showing by combining the above arguments and the following “translation invariance” of the discrepancy. More precisely, let $\xi$ be a constant and $\xi_{n}$ be a sequence of real number. Let $D_{\xi,N}$ be the discrepancy of the fractional parts

[TABLE]

Thus $D_{N}=D_{0,N}$. From the definition of discrepancy (2.6) we derive

[TABLE]

Easy calculations, show that Theorem 2.14 with $k=1$, $\varphi_{1}(T)=T^{d}$ and thus with $\sigma_{1}(\bm{\varphi})=d(d-1)/2$, implies the result.

6. Comments

6.1. Discrepancy of polynomials

First of all we give a proof for (2.7), that is, that for almost all $\mathbf{u}\in\mathsf{T}_{d}$ one has

[TABLE]

This is based on [15, Theorem 5.13] (see Proposition 6.2 below) and the following general statement which is perhaps well-known but the authors have not been able to find it in the literature.

Proposition 6.1**.**

Let ${\mathcal{U}}\subseteq\mathsf{T}_{d}$, be a set of positive Lebesgue measure $\lambda({\mathcal{U}})>0$. Then there is a vector $\mathbf{v}_{0}\in\mathsf{T}_{d}$ and a set real numbers ${\mathcal{W}}\subseteq[0,\sqrt{d}]$ of positive Lebesgue measure $\lambda({\mathcal{W}})>0$, such that for every $w\in{\mathcal{W}}$ we have $w\mathbf{v}_{0}\in{\mathcal{U}}$.

Proof.

Let $\chi_{{\mathcal{U}}}$ be the characteristic function of ${\mathcal{U}}$, then clearly we have

[TABLE]

Applying the polar coordinates [12, Theorem 3.12] to the function $\chi_{{\mathcal{U}}}$, we obtain

[TABLE]

where $\mathcal{H}^{d-1}$ is the $(d-1)$-dimensional Hausdorff measure which is given by [12, Chapter 2]. By taking $\mathbf{u}=r\mathbf{v}$ for some $\mathbf{v}\in\mathbb{S}^{d-1}$ in the second term of (6.2) we obtain

[TABLE]

where $\mathbb{S}^{d-1}\subseteq{\mathbb{R}}^{d}$ is the unit sphere centred at the origin. Combining this with (6.1) and (6.2) and applying Fubini’s theorem, we arrive to

[TABLE]

Since $\lambda({\mathcal{U}})>0$, we conclude that there exist a vector

[TABLE]

and a set of $r\in[0,\sqrt{d}]$ of positive Lebesgue measure such that $r\mathbf{v}\in{\mathcal{U}}$, which gives the desired result.     \sqcap$$\sqcup

We formulate [15, Theorem 5.13], see also [2], in the following form.

Proposition 6.2**.**

Let ${\mathcal{A}}=\left(a_{n}\right)_{n=1}^{\infty}$ be a sequence increasing sequence of real numbers such that $a_{n+1}-a_{n}\geqslant\delta>0$ and let $\varepsilon>0$. Then for almost all $w\in{\mathbb{R}}$ we have

[TABLE]

where $D\left(w{\mathcal{A}};N\right)$ means the discrepancy of the sequence $wa_{n}\pmod{1}$, $n=1,\ldots,N$.

Let us now fix some $\varepsilon>0$ and denote by ${\mathcal{U}}\subseteq\mathsf{T}_{d}$ the set of $\mathbf{u}\in\mathsf{T}_{d}$, for which

[TABLE]

for infinitely many $N\in{\mathbb{N}}$. Assume that $\lambda({\mathcal{U}})>0$. By Proposition 6.1 there exists a vector

[TABLE]

and a set of $w\in[0,\sqrt{d}]$ of positive Lebesgue measure such that we have (6.3) with $\mathbf{u}=w\mathbf{v}$ and for infinitely many $N\in{\mathbb{N}}$. On the other hand applying Proposition 6.2 to the sequence $v_{1}n+\ldots+v_{d}n^{d}$, $n\in{\mathbb{N}}$ and parameter $\varepsilon/2$, we obtain that for almost all $w\in{\mathbb{R}}$ one has

[TABLE]

This now gives the contradiction and therefore together with the arbitrary choice of $\varepsilon$ the estimate (2.7) holds.

At the moment we are not able to rule out that for almost all $\mathbf{u}\in\mathsf{T}_{d}$ one has

[TABLE]

for any $d\geqslant 3$, which we believe to be false. In fact, as we have mentioned, we believe that (2.7) is tight except the logarithm factor, and this is true for the case $d=2$ which follows from a result of Fedotov and Klopp [13, Theorem 0.1] and the Koksma inequality [15, Theorem 5.4], while the conjecture is still open when $d\geqslant 3$.

For the monomial sequence $xn^{d}$, $n\in{\mathbb{N}}$ we denote by $D_{d}(x;N)$ the corresponding discrepancy of its fractional parts. We note that in the case $d=1$ the celebrated result of Khintchine, see [11, Theorem 1.72], implies that for almost all $x\in[0,1)$ one has

[TABLE]

Finally, we remark that for $d\geqslant 2$ Aistleitner and Larcher [1, Corollary 1] have recently shown that for almost all $x\in[0,1)$ one has

[TABLE]

for any $\varepsilon>0$ and for infinitely many $N\in{\mathbb{N}}$. Many other metrical results on the discrepancy of polynomials and other sequences can be found in [1, 3, 15].

6.2. The structure of the exceptional sets

For $1\leqslant k<d$ and $0<\alpha<1$ denote

[TABLE]

Theorem 2.1 claims that for any

[TABLE]

the set ${\mathcal{E}}_{\bm{\varphi},k,\alpha}$ is of zero $k$-dimensional Lebesgue measure. It is natural to ask what we can say more for these sets with zero Lebesgue measure.

Motivated from the works [8, 9] we ask the size of ${\mathcal{E}}_{\bm{\varphi},k,\alpha}$ in the sense of Baire categories and Hausdorff dimension. In the following suppose that $\bm{\varphi}$ is the classical choice as in (1.6). The argument in [8] implies that for any $1\leqslant k<d$ and any $0<\alpha<1$ the set

[TABLE]

is of first Baire category in $\mathsf{T}_{k}$. For the Hausdorff dimension, [9, Corollary 1.3] implies that

[TABLE]

where $\dim$ means the Hausdorff dimension. We omit these details here.

6.3. Further possible extensions

We have formulated Theorems 2.1 and 2.3 in terms of the unit torus $\mathsf{T}_{d}$ only. In fact these result may shed some light for subsets of $\mathsf{T}_{d}$ also. For instance, Theorem 2.1 implies the following statement.

Let ${\mathcal{A}}\subseteq\mathsf{T}_{d}$ such that $\lambda(\pi_{d,k}({\mathcal{A}}))>0,$ where the notation $\pi_{d,k}$ is given by (2.2) and the symbol $\lambda$ represents the $k$-dimensional Lebesgue measure. Note that the set ${\mathcal{A}}$ itself may be of vanishing $d$-dimensional Lebesgue measure. Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically, then for almost all $\mathbf{x}\in\pi_{d,k}({\mathcal{A}})$ one has

[TABLE]

where $\Gamma(\bm{\varphi},k)$ comes from Theorem 2.1.

Indeed, let ${\mathcal{G}}\subseteq\mathsf{T}_{d}$ be the collection of vectors $\mathbf{x}\in\mathsf{T}_{k}$ which satisfy (6.4), then Theorem 2.1 implies that the set ${\mathcal{G}}$ has full measure (that is, $\lambda({\mathcal{G}})=1$), and therefore

[TABLE]

Thus the bound (6.4) holds for almost all $\mathbf{x}\in\pi_{d,k}({\mathcal{A}})$.

Now, suppose that $\lambda(\pi_{d,k}({\mathcal{A}}))=0$ for some subset ${\mathcal{A}}\subseteq\mathsf{T}_{d}$. It is interesting to investigate whether one can obtain an appropriate analogue of Theorem 2.1 in this case. In the following we formulate a general framework for the possible extension of Theorem 2.1.

Let ${\mathcal{A}}\subseteq\mathsf{T}_{d}$ and $\mu$ be a “nice” probability measure on ${\mathcal{A}}$, for example a Borel measure. Suppose that the measure $\mu$ admits some kind of the mean value theorem, that is, there exist positive constants $s<t$ such that ($a_{n}=n^{o(1)}$)

[TABLE]

Furthermore, assume that $\mu$ has some regular properties, for instance, there exist positive constants $\beta_{1}<\beta_{2}$ such that for any ball ${\mathcal{B}}(\mathbf{u},r)$ centred at $\mathbf{u}\in{\mathcal{A}}$ and of positive radius $r<1$ one has

[TABLE]

with some absolute implied constants. Note that the condition (6.6) gives the upper bound and lower bound on the measure of any given high dimensional rectangle.

We remark that our methods work for any subset ${\mathcal{A}}\subseteq\mathsf{T}_{d}$ and any measure $\mu$ on ${\mathcal{A}}$ which has the above properties (6.5) and (6.6). More precisely, let $\mu_{d,k}$ be the projection measure of $\pi_{d,k}$, that is,

[TABLE]

Suppose that $\bm{\varphi}\in{\mathbb{Z}}[T]^{d}$ is such that the Wronskian $W(T;\bm{\varphi})$ does not vanish identically, then for $\mu_{d,k}$-almost all $\mathbf{x}\in\pi_{d,k}({\mathcal{A}})$ one has

[TABLE]

where $\Gamma({\mathcal{A}},\mu,\bm{\varphi})$ is a positive constant which can be explicitly evaluated in terms of the parameters in (6.5) and (6.6). We expect that

[TABLE]

holds in many natural situations.

On the other hand, it is not clear how to extend Theorem 2.14 (the result of the discrepancy) to subsets ${\mathcal{A}}\subseteq\mathsf{T}_{d}$ with some measure $\mu$ on ${\mathcal{A}}$ since in general $\mu$ does not have the invariant property as in Lemma 5.2. For example, the famous $\times 2\times 3$-Conjecture of Furstenberg, which still remains open (see [18, 24] and references therein): using our notation as in Lemma 5.2, let $\mu$ be a Borel probability measure on $[0,1)$ such that for any “nice” subset ${\mathcal{F}}\subseteq[0,1)$ the identity

[TABLE]

holds for $g=2$ and $g=3$, then $\mu$ is Lebesgue measure or some “trivial” measure.

Acknowledgement

The authors are grateful to Trevor Wooley for helpful discussions and patient answering their questions. The authors also would like to thank the anonymous referee for the very careful reading of the manuscript and many helpful comments.

This work was supported by ARC Grant DP170100786.