High-dimensional Lehmer problem on Beatty sequences

Xiaoqing Zhao; Yuan Yi; Xiaoqing Zhao; Yuan Yi

doi:10.3934/math.2023684

AIMS Mathematics

2023, Volume 8, Issue 6: 13492-13502. doi: 10.3934/math.2023684

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Research article

High-dimensional Lehmer problem on Beatty sequences

Xiaoqing Zhao ,
Yuan Yi ^,

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received: 25 December 2022 Revised: 18 February 2023 Accepted: 09 March 2023 Published: 07 April 2023
MSC : 11B83, 11L05, 11N69

Let $q$ be a positive integer. For each integer $a$ with $1 \leqslant a < q$ and $(a, q) = 1$ , it is clear that there exists one and only one $\bar{a}$ with $1 \leqslant\bar{a} < q$ such that $a \bar{a} \equiv 1(q)$ . Let $k$ be any fixed integer with $k \geq 2, 0 < \delta_{i} \leq 1, i = 1, 2, \cdots, k.$ $r_{n}\left(\delta_{1}, \delta_{2}, \cdots, \delta_{k}, \alpha, \beta, c; q\right)$ denotes the number of all $k$ -tuples with positive integer coordinates $\left(x_{1}, x_{2}, \ldots, x_{k}\right)$ such that $1 \leq x_{i} \leq \delta_{i}q, \left(x_{i}, q\right) = 1, x_{1} x_{2} \cdots x_{k} \equiv c(q)$ , and $x_{1}, x_{2}, \cdots, x_{k-1} \in B_{\alpha, \beta}$ . In this paper, we consider the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals and give an asymptotic formula by the properties of Beatty sequences and the estimates for hyper Kloosterman sums.

Keywords:

Citation: Xiaoqing Zhao, Yuan Yi. High-dimensional Lehmer problem on Beatty sequences[J]. AIMS Mathematics, 2023, 8(6): 13492-13502. doi: 10.3934/math.2023684

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Abstract

1. Introduction

Let $q$ be a positive integer. For each integer $a$ with $1 \leqslant a < q, (a, q) = 1$ , we know that there exists one and only one $\bar{a}$ with $1 \leqslant\bar{a} < q$ such that $a \bar{a} \equiv 1(q)$ . Let $r(q)$ be the number of integers $a$ with $1 \leqslant a < q$ for which $a$ and $\bar{a}$ are of opposite parity.

D. H. Lehmer (see ^[1]) posed the problem to investigate a nontrivial estimation for $r(q)$ when $q$ is an odd prime. Zhang ^[2,3] gave some asymptotic formulas for $r(q)$ , one of which reads as follows:

$r(q) = \frac{1}{2} \phi(q)+O\left(q^{\frac{1}{2}} d^{2}(q) \log ^{2} q\right).$

Zhang ^[4] generalized the problem over short intervals and proved that

$\sum\limits_{\substack{a \leq N \\ a \in R(q)}} 1 = \frac{1}{2} N \phi(q) q^{-1}+O\left(q^{\frac{1}{2}} d^{2}(q) \log ^{2} q\right),$

where

$R(q): = \{a: 1 \leqslant a \leqslant q,(a, q) = 1,2 \nmid a+\bar{a}\} .$

Let $n \geqslant 2$ be a fixed positive integer, $q \geqslant 3$ and $c$ be two integers with $(n, q) = (c, q) = 1$ . Let $0 < \delta_{1}, \delta_{2} \leq 1.$ Lu and Yi ^[5] studied the Lehmer problem in the sense of short intervals as

$r_{n}\left(\delta_{1}, \delta_{2}, c ; q\right): = \mathop {\sum\limits_{{a \leqslant \delta_{1} q}} {\sum\limits_{{\bar{a} \leqslant \delta_{2} q}} {} } }\limits_{\scriptstyle{a \bar{a}\equiv c\bmod q}\atop \scriptstyle {n \nmid a+\bar{a} }} 1 ,$

and obtained an interesting asymptotic formula,

$r_{n}\left(\delta_{1}, \delta_{2}, c ; q\right) = \left(1-n^{-1}\right) \delta_{1} \delta_{2} \phi(q)+O\left(q^{\frac{1}{2}} d^{6}(q) \log ^{2} q\right).$

Liu and Zhang ^[6] $r$ -th residues and roots, and obtained two interesting mean value formulas. Guo and Yi ^[7] found the Lehmer problem also has good distribution properties on Beatty sequences. For fixed real numbers $\alpha$ and $\beta$ , the associated non-homogeneous Beatty sequence is the sequence of integers defined by

$\mathcal{B}_{\alpha, \beta}: = (\lfloor\alpha n+\beta\rfloor)_{n = 1}^{\infty},$

where $\lfloor t\rfloor$ denotes the integer part of any $t \in \mathbb{R}$ . Such sequences are also called generalized arithmetic progressions. If $\alpha$ is irrational, it follows from a classical exponential sum estimate of Vinogradov ^[8] that $\mathcal{B}_{\alpha, \beta}$ contains infinitely many prime numbers; in fact, one has the asymptotic estimate

$\#\left\{\text { prime } p \leqslant x: p \in \mathcal{B}_{\alpha, \beta}\right\} \sim \alpha^{-1} \pi(x) \quad \text { as } \quad x \rightarrow \infty$

where $\pi(x)$ is the prime counting function.

We define type $\tau = \tau(\alpha)$ for any irrational number $\alpha$ by the following definition:

$\tau: = \sup \left\{t \in \mathbb{R}: \liminf \limits_{n \rightarrow \infty} n^t\|\alpha n\| = 0\right\}.$

Based on the results obtained, we consider the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals in this paper. That is,

$r_{n}\left(\delta_{1}, \delta_{2}, \cdots ,\delta_{k}, c, \alpha, \beta ; q\right): = \mathop {\sum\limits_{{x_{1} \leqslant \delta_{1} q}} { \cdots \sum\limits_{{x_{k} \leqslant \delta_{k} q}} {} } }\limits_{\scriptstyle {x_{1} \cdots x_{k} \equiv c\bmod q } \atop {\scriptstyle {x_{1}, \cdots x_{k-1} \in B_{\alpha,\beta}} \atop \scriptstyle {n \nmid x_{1}+\cdots+x_{k}}}} 1,(0 < \delta_{1}, \delta_{2},\cdots, \delta_{k} \leq 1),$

and where k = 2, we get the result of ^[7].

By using the properties of Beatty sequences and the estimates for hyper Kloosterman sums, we obtain the following result.

Theorem 1.1. Let $k \geq 2$ be a fixed positive integer, $q\geq n^{3}$ and $c$ be two integers with $(n, q) = (c, q) = 1$ , and $\delta_{1}, \delta_{2}, \cdots, \delta_{k}$ be real numbers satisfying $0 < \delta_{1}, \delta_{2}, \cdots, \delta_{k} \leq 1$ . Let $\alpha > 1$ be an irrational number of finite type. Then, we have the following asymptotic formula:

$r_{n}\left(\delta_{1}, \delta_{2}, \cdots ,\delta_{k}, c, \alpha, \beta ; q\right) = \left(1-n^{-1}\right) \alpha^{-(k-1)} \delta_{1} \delta_{2} \cdots \delta_{k}\phi^{k-1}(q)+O(q^{k-1-\frac{1}{\tau+1}+\varepsilon} ),$

where $\phi(\cdot)$ is the Euler function, $\varepsilon$ is a sufficiently small positive number, and the implied constant only depends on $n$ .

Notation. In this paper, we denote by $\lfloor t\rfloor$ and $\{t\}$ the integral part and the fractional part of $t$ , respectively. As is customary, we put

$\mathbf{e}(t): = e^{2 \pi i t} \quad \text { and } \quad\{t\}: = t-\lfloor t\rfloor .$

The notation $\|t\|$ is used to denote the distance from the real number $t$ to the nearest integer; that is,

$\|t\|: = \min \limits_{n \in \mathbb{Z}}|t-n| .$

Let $\chi^{0}$ be the principal character modulo $q$ . The letter $p$ always denotes a prime. Throughout the paper, $\varepsilon$ always denotes an arbitrarily small positive constant, which may not be the same at different occurrences; the implied constants in symbols $O, \ll$ and $\gg$ may depend (where obvious) on the parameters $\alpha, n, \varepsilon$ but are absolute otherwise. For given functions $F$ and $G$ , the notations $F \ll G$ , $G \gg F$ and $F = O(G)$ are all equivalent to the statement that the inequality $|F| \leqslant \mathcal{C}|G|$ holds with some constant $\mathcal{C} > 0$ .

2. Preliminary lemmas

To complete the proof of the theorem, we need the following several definitions and lemmas.

Definition 2.1. For an arbitrary set $\mathcal{S}$ , we use $\mathbf{1}_{\mathcal{S}}$ to denote its indicator function:

$\mathbf{1}_{\mathcal{S}}(n): = \begin{cases}1 & { if } \;n \in \mathcal{S}, \\ 0 & { if }\; n \notin \mathcal{S} .\end{cases}$

We use $\mathbf{1}_{\alpha, \beta}$ to denote the characteristic function of numbers in a Beatty sequence:

$\mathbf{1}_{\alpha, \beta}(n): = \begin{cases}1 & { if } \;n \in \mathcal{B}_{\alpha, \beta}, \\ 0 & { if }\; n \notin \mathcal{B}_{\alpha, \beta}.\end{cases}$

Lemma 2.2. Let $a, q$ be integers, $\delta \in(0, 1)$ be a real number, $\theta$ be a rational number. Let $\alpha$ be an irrational number of finite type $\tau$ and $H = q^{\varepsilon} > 0$ . We have

$\sum\limits_{\scriptstyle {a \le \delta q} \atop \scriptstyle{a \in {{\cal B}_{\alpha ,\beta }}}} ' 1 = \alpha^{-1} \delta \phi(q)+O\left((\phi(q))^{\frac{\tau}{\tau+1}+\varepsilon}\right),$

and

$\sum\limits_{\substack{a \leqslant \delta q \\ a \in \mathcal{B}_{\alpha, \beta}}} \mathbf{e}(\theta a) = \alpha^{-1} \sum\limits_{a \leqslant \delta_1 q} \mathbf{e}(\theta a)+O\left(\|\theta\|^{-1} q^{-\varepsilon}+q^{\varepsilon}\right).$

Taking

$H = \|\theta\|^{-\frac{1}{\tau+1}+\varepsilon},$

we have

$\sum\limits_{\substack{a \leqslant \delta q \\ a \in B_{\alpha, \beta}}} \mathbf{e}(\theta a) = \alpha^{-1} \sum\limits_{a \leqslant \delta_1 q} \mathbf{e}(\theta a)+O\left(\|\theta\|^{-\left(\frac{\tau}{\tau+1}+\varepsilon\right)}\right) .$

Proof. This is Lemma 2.4 and Lemma 2.5 of ^[7].

Lemma 2.3. Let

$\mathbf{Kl}(r_{1},r_{2},\cdots,r_{k};q) = \sum\limits_{x_{1} \leqslant q-1} \cdots \sum\limits_{x_{k-1} \leqslant q-1} \mathbf{e}\left(\frac {r_{1}x_{1}+\cdots+r_{k-1}x_{k-1}+ r_{k}\overline{x_{1} \cdots x_{k-1}}}{p}\right).$

Then

$\mathbf{Kl}(r_{1},r_{2},\cdots,r_{k};q) \ll q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, r_{k}, q\right)^{\frac{1}{2}} \cdots\left(r_{k-1}, r_{k}, q\right)^{\frac{1}{2}}$

where $(a, b, c)$ is the greatest common divisor of $a, b$ and $c$ .

Proof. See ^[9].

Lemma 2.4. Assume that $U$ is a positive real number, $K$ is a positive integer and that $a$ and $b$ are two real numbers. If

$a = \frac{s}{r}+\frac{\theta}{r^{2}}, \quad(r, s) = 1, r \geq 1,|\theta| \leq 1,$

then

$\sum\limits_{k \leqslant K} \min (U, \frac{1}{\|a k+b\|}) \ll (\frac{K}{r}+1 )(U+r \log r).$

Proof. The proof is given in ^[10].

3. Proof of theorem

We begin by the definition

$r_{n}\left(\delta_{1}, \delta_{2}, \cdots ,\delta_{k}, c, \alpha, \beta ; q\right) = S_{1}-S_{2},$

where

$S_{1}: = \mathop {\sum\limits_{{x_{1} \leqslant \delta_{1} q}} { \cdots \sum\limits_{{x_{k} \leqslant \delta_{k} q}} {} } }\limits_{\scriptstyle {x_{1} \cdots x_{k} \equiv c\bmod q } \atop {\scriptstyle {x_{1}, \cdots x_{k-1} \in \mathcal{B}_{\alpha,\beta}} }} 1,$

and

$S_{2}: = \mathop {\sum\limits_{{x_{1} \leqslant \delta_{1} q}} { \cdots \sum\limits_{{x_{k} \leqslant \delta_{k} q}} {} } }\limits_{\scriptstyle{x_{1} \cdots x_{k} \equiv c\bmod q }\atop {\scriptstyle{x_{1}, \cdots x_{k-1} \in \mathcal{B}_{\alpha,\beta}}\atop \scriptstyle{n \mid x_{1}+\cdots+x_{k}}}} 1.$

By the Definition 2.1, Lemma 2.2 and congruence properties, we have

$\begin{aligned} S_{1}& = \mathop{\sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q}}_{x_{1} \cdots x_{k} \equiv c\bmod q }\mathbf{1}_{\alpha,\beta}\left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\\ & = \frac{1}{\phi(q)} \sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q} \sum\limits_{\chi \bmod q}\chi(x_{1}) \cdots \chi(x_{k}) \chi(\overline c)\mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\\ & = S_{11}+S_{12}, \end{aligned}$

where

$\begin{align*} S_{11}: = \frac{1}{\phi(q)}\mathop{ {\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \cdots \mathop{ {\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q} \mathbf{1}_{\alpha,\beta}\left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right), \end{align*}$

and

$S_{12}: = \frac{1}{\phi(q)} \mathop{\sum\limits_{\chi \bmod q}}_{\chi \neq \chi_{0}} \chi(\overline c) \left(\sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q} \chi(x_{1}) \cdots \chi(x_{k}) \mathbf{1}_{\alpha,\beta} ( x_{1}) \cdots \mathbf{1}_{\alpha,\beta}( x_{k-1} )\right).$

For $S_{2}$ , it follows that

$\begin{aligned} S_{2}& = \frac{1}{\phi(q)} \mathop{\sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q}}_{n \mid x_{1}+\cdots+x_{k}} \sum\limits_{\chi \bmod q}\chi(x_{1}) \cdots \chi(x_{k}) \chi(\overline c)\mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\\ & = S_{21}+S_{22}, \end{aligned}$

where

$\begin{align*} S_{21}: = \frac{1}{\phi(q)} \mathop{\mathop{ {\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \cdots \mathop{ {\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q} }_{n \mid x_{1}+\cdots+x_{k}} \mathbf{1}_{\alpha,\beta}\left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right), \end{align*}$

and

$\begin{align*} S_{22}: = \frac{1}{\phi(q)} \mathop{\sum\limits_{\chi \bmod q}}_{\chi \neq \chi_{0}} \chi(\overline c) \mathop{{\sum\limits_{x_{1} \leqslant \delta_{1} q}} \cdots {\sum\limits_{x_{k} \leqslant \delta_{k} q}}}_{n \mid x_{1}+\cdots+x_{k}} \chi(x_{1}) \cdots \chi(x_{k-1}) \mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1}\right) . \end{align*}$

3.1. Estimation of $S_{11}$

From the classical bound

$\sum \limits_{a \le \delta q}' 1 = \delta \phi(q)+O\left(d(q)\right)$

and Lemma 2.2, we have

$\begin{align} S_{11}& = \frac{1}{\phi(q)} \left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right) \left(\mathop{{\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q}1\right) \\ & = \left(\delta_{k}+O\left(\frac{d(q)}{\phi(q)}\right)\right)\prod \limits_{i = 1}^{k-1}\left( \alpha^{-1} \delta_{i} \phi(q)+O\left((\phi(q))^{\frac{\tau}{\tau+1}+\varepsilon}\right)\right) \\ & = \alpha^{-(k-1)}\phi^{k-1}(q)\prod \limits_{i = 1}^{k-1} \delta_{i}+O\left(q^{k-1-\frac{1}{\tau+1}+\varepsilon}\right). \end{align}$

(3.1)

3.2. Estimation of $S_{21}$

From Lemma 2.2, we obtain

$\begin{align} S_{21}& = \frac{1}{\phi(q)}\left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right)\left(\mathop{\mathop{{\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q}}_{n \mid x_{k}+(x_{1}+ \cdots +x_{k-1})}1\right) \\ & = \frac{1}{\phi(q)}\left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right) \left(\mathop{\sum\limits_{x_{k} \leqslant \delta_{k} q }}_{x_{k} \equiv-(x_{1}+ \cdots +x_{k-1}) \bmod n} \sum\limits_{\substack{d \mid(x_{k}, q)}} \mu(d)\right)\\ & = \frac{1}{\phi(q)}\left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right)\left( \sum\limits_{\substack{d \mid q}} \mu(d) \mathop{\mathop{\sum\limits_{x_{k} \leqslant \delta_{k}q}}_{d \mid x_{k}}}_{x_{k} \equiv-(x_{1}+ \cdots +x_{k-1})\bmod n} 1 \right) \\ & = \frac{1}{\phi(q)} \left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right)\left( \sum\limits_{\substack{d \mid q}} \mu(d) \left( \frac{\delta_{k}q}{nd}+O(1)\right)\right) \\ & = \frac{1}{\phi(q)}\left(\frac{\delta_{k}\phi(q)}{n}+O\left(d(q)\right) \right)\prod \limits_{i = 1}^{k-1}\left( \alpha^{-1} \delta_{i} \phi(q)+O\left((\phi(q))^{\frac{\tau}{\tau+1}+\varepsilon}\right)\right)\\ & = \alpha^{-(k-1)}n^{-1}\phi^{k-1}(q)\prod \limits_{i = 1}^{k-1} \delta_{i}+O (q^{k-1-\frac{1}{\tau+1}+\varepsilon} ). \end{align}$

(3.2)

3.3. Estimation of $S_{22}$ and $S_{12}$

By the properties of exponential sums,

$\begin{align} S_{22} = &\frac{1}{n \phi(q)} \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c) \left({\sum\limits_{x_{1} \leqslant \delta_{1} q}}\cdots {\sum\limits_{x_{k} \leqslant \delta_{k-1} q}}\chi(x_{1}) \cdots \chi(x_{k}) \mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\right) \\ &\times \left(\sum \limits_{l = 1}^{n}\mathbf{e}(\frac{x_{1}+\cdots+x_{k}}{n}l) \right)\\ = &\frac{1}{n \phi(q)} \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c)\sum \limits_{l = 1}^{n} \prod \limits_{i = 1}^{k-1}\left( \sum\limits_{x_{i} \leqslant \delta_{i} q}\mathbf{1}_{\alpha, \beta}(x_{i}) \chi(x_{i}) \mathbf{e}(\frac{x_{i}}{n} l)\right)\left( \sum\limits_{x_{k} \leqslant \delta_{k} q} \chi(x_{k}) \mathbf{e} (\frac{x_{k}}{n} l)\right). \end{align}$

(3.3)

Let

$G(r, \chi): = \sum\limits_{h = 1}^{q} \chi(h) \mathbf{e} (\frac{r h}{q} )$

be the Gauss sum, and we know that for $\chi \neq \chi_{0}$ ,

$\chi(x_{i}) = \frac{1}{q} \sum\limits_{r = 1}^{q} G(r, \chi) \mathbf{e} (-\frac{x_{i} r}{q} ) = \frac{1}{q} \sum\limits_{r = 1}^{q-1} G(r, \chi) \mathbf{e} (-\frac{x_{i} r}{q} ),$

and

$\frac{l}{n}-\frac{r}{q} \neq 0$

for $1 \leqslant l \leqslant n, 1 \leqslant r \leqslant q-1$ and $(n, q) = 1$ .

Therefore,

$\begin{equation} \sum\limits_{x_{k} \leqslant \delta_{k} q} \chi(x_{k}) \mathbf{e} (\frac{x_{k}}{n} l ) = \frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e} (\frac{r_{k}}{q}-\frac{l}{h})-1}, \end{equation}$

(3.4)

where

$f(\delta, l, r ; n, p): = 1-\mathbf{e}\left( (\frac{l}{n}-\frac{r}{q} )\lfloor\delta q\rfloor\right)$

and

$\left|f\left(\delta_{k}, l, r_{k} ; n, q\right)\right| \leqslant 2.$

For $x_{i}(1\leqslant i \leqslant k-1)$ , using Lemma 2.2, we also have

$\begin{align} & \sum\limits_{x_{i} \leqslant \delta_{i} q} \mathbf{1}_{\alpha, \beta}(x_{i}) \chi(x_{i}) \mathbf{e} (\frac{x_{i}}{n} l ) \\ = & \frac{1}{q} \sum\limits_{x_{i} \leqslant \delta_{i} q} \mathbf{1}_{\alpha, \beta}(x_{i}) \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \mathbf{e}\left( (\frac{l}{n}-\frac{r_{i}}{q} ) x_{i}\right) \\ = & \frac{1}{q} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \sum\limits_{x_{i} \leqslant \delta_{i} q} \mathbf{1}_{\alpha, \beta}(x_{i}) \mathbf{e}\left( (\frac{l}{n}-\frac{r_{i}}{q} ) x_{i}\right) \\ = & \frac{1}{q} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \left(\alpha^{-1}\sum\limits_{a \leqslant \delta_{i} q} \mathbf{e}\left( (\frac{l}{n}-\frac{r_{i}}{q} ) x_{i}\right)+O\left(\frac{q^{-\varepsilon}}{\|\frac{l}{n}-\frac{r_{i}}{q}\|} +q^{\varepsilon}\right)\right) \\ = &\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \left(\frac{f\left(\delta_{i} , l, r_{i} ; n, q\right)}{\mathbf{e} (\frac{r_{i}}{q}-\frac{l}{n} )-1}+O\left(\frac{q^{-\varepsilon}}{\|\frac{l}{n}-\frac{r_{i}}{q}\|} +q^{\varepsilon}\right)\right) . \end{align}$

(3.5)

Let

$\begin{align} S_{23}& = \frac{1}{n \phi(q)} \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c)\sum \limits_{l = 1}^{n}\prod\limits_{i = 1}^{k-1} \left(\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \frac{f\left(\delta_{i}, l, r_{i} ; n, q\right)}{\mathbf{e} (\frac{r_{i}}{q}-\frac{l}{n})-1}\right)\left(\frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e}(\frac{r_{k}}{q}-\frac{l}{n})-1}\right) \\ & = \frac{1}{n \phi(q) q^{k} \alpha^{k-1}} \sum\limits_{l = 1}^{n}\sum\limits_{r_{1} = 1}^{q-1}\cdots \sum\limits_{r_{k} = 1}^{q-1} \frac{f\left(\delta_{1} , l, r_{1} ; n, q\right)\cdots f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\left(\mathbf{e} (\frac{r_{1}}{q}-\frac{l}{n} )-1\right)\cdots \left(\mathbf{e} (\frac{r_{k}}{q}-\frac{l}{n} )-1\right)} \\ &\times \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right). \end{align}$

(3.6)

From the definition of Gauss sum and Lemma 2.3, we know that

$\begin{align} &\sum\limits_{\chi \mathrm{mod} q}\chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right)\\ = &\sum\limits_{h_{1} = 1}^{q-1}\cdots \sum\limits_{h_{k} = 1}^{q-1}\sum\limits_{\chi \mathrm{mod}q}\chi(\overline c)\chi(h_{1})\cdots \chi(h_{k})\mathbf{e} ( \frac{r_{1}h_{1}+\cdots +r_{k}h_{k}}{q} )\\ = &\phi(q)\mathop{\sum\limits_{h_{1} = 1}^{q-1}\cdots \sum\limits_{h_{k} = 1}^{q-1}}_{h_{1} \cdots h_{k} \equiv c \bmod q}\mathbf{e} ( \frac{r_{1}h_{1}+ \cdots +r_{k}h_{k}}{q} )\\ = &\phi(q)\sum\limits_{h_{1} = 1}^{q-1}\cdots \sum\limits_{h_{k} = 1}^{q-1}\mathbf{e} ( \frac{r_{1}h_{1}+ \cdots r_{k-1}h_{k-1}+r_{k}c\overline{h_{1} \cdots h_{k-1}}}{q} )\\ = &\phi(q) \mathbf{Kl}(r_{1},r_{2},\cdots,r_{k}c;q) \\ \ll& \phi(q) q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, r_{k}c, q\right)^{\frac{1}{2}} \cdots\left(r_{k-1}, r_{k}c, q\right)^{\frac{1}{2}} \\ \ll&\phi(q) q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, q\right) \cdots\left(r_{k}, q\right). \end{align}$

(3.7)

By Mobius inversion, we get

$G(r, \chi_{0}) = \sum\limits_{h = 1}^{q}' \mathbf{e} (\frac{r h}{q} ) = \mu\left(\frac{q}{(r, q)}\right) \frac{\varphi(q)}{\varphi(q /(r, q))} \ll(r, q),$

and

$\chi_{0}(\overline c)G\left(r_{1}, \chi_{0}\right)\cdots G\left(r_{k}, \chi_{0}\right) \ll\left(r_{1}, q\right) \cdots\left(r_{k}, q\right).$

Hence,

$\begin{align} &\mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}}\chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right)\\ = &\sum\limits_{\chi \mathrm{mod} q}\chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right)-\chi_{0}(\overline c)G\left(r_{1}, \chi_{0}\right)\cdots G\left(r_{k}, \chi_{0}\right)\\ \ll&\phi(q) q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, q\right) \cdots\left(r_{k}, q\right). \end{align}$

(3.8)

From (3.8) we may deduce the following result:

$\begin{align} S_{23}&\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{r = 1}^{q-1} \frac{(r,q)}{\left|\mathbf{e} (\frac{r}{q}-\frac{l}{n} )-1\right|}\right)^{k}\\ &\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{r = 1}^{q-1} \frac{(r,q)}{\left|\sin \pi (\frac{r}{q}-\frac{l}{n} )\right|}\right)^{k}\\ &\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{r = 1}^{q-1} \frac{(r,q)}{\|\frac{r}{q}-\frac{l}{n}\|}\right)^{k}\\ & = \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q}}_{d < q }\mathop{\sum\limits_{r \leq q-1}}_{(r,q) = d }\frac{d}{\|\frac{r}{q}-\frac{l}{n}\|}\right)^{k }\\ & = \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q}}_{d < q }d\mathop{\sum\limits_{m \leq\frac{q-1}{d} }}_{(m,q) = 1}\frac{1}{\|\frac{md}{q}-\frac{l}{n}\|}\right)^{k }\\ & = \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q}}_{d < q }d\sum\limits_{k \mid q}\mu(k)\sum\limits_{m \leq\frac{q-1}{kd} }\frac{1}{\|\frac{mkd}{q}-\frac{l}{n}\|}\right)^{k }. \end{align}$

It is easy to see

$\|\frac{mkd}{q}-\frac{l}{n}\| = \|\frac{mkn-l(q/d)}{(q/d)n}\| \geq \frac{1}{(q/d)n},$

and we obtain

$S_{23}\ll\frac{k^{\omega(q)}}{n \phi(q) q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q\\{d < q }}}d\sum\limits_{k \mid q}\sum\limits_{m \leq\frac{q-1}{kd} }\min (\frac{qn}{d},\frac{1}{\|\frac{mkd}{q}-\frac{l}{n}\|} )\right)^{k }.$

Let $k d / q = h_{0} / q_{0}$ , where $q_{0} \geq 1, \left(h_{0}, q_{0}\right) = 1$ , and we will easily obtain $q /(k d) \leq q_{0} \leq q / d$ . By using Lemma 2.4, we have

$\begin{align} S_{23}&\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{\substack{d \mid q \\ d < q}} d \sum\limits_{k \mid q}\left(\frac{(q-1) /(k d)}{q_{0}}+1\right) (\frac{q n}{d}+q_{0} \log q_{0} )\right)^{k}\\ &\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{\substack{d \mid q \\ d < q}} d \sum\limits_{k \mid q}\left(\frac{(q-1) /(k d)}{q/(kd)}+1\right) (\frac{q n}{d}+\frac{q}{d} \log \frac{q}{d} )\right)^{k}\\ &\ll \frac{k^{\omega(q)}q^{\frac{k-1}{2}}}{ \alpha^{k-1}}\left(\sum\limits_{\substack{d \mid q \\ d < q}} \sum\limits_{k \mid q}n+\log q\right)^{k}\\ &\ll q^{\frac{k-1}{2}}d^{2k}(q)(\log q+n)^{k}. \end{align}$

Let

$S_{24}: = \frac{q^{(k-1)(-\varepsilon)}}{n \phi(q)}\mathop{\sum\limits_{\chi \mathrm{mod} q\\{\chi \neq \chi_{0}}} }\chi(\overline c)\sum \limits_{l = 1}^{n}\prod\limits_{i = 1}^{k-1}\left(\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G(r_{i}, \chi)\frac{1}{\|\frac{l}{n}-\frac{r_{i}}{q}\|} \right) \left(\frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e}(\frac{r_{k}}{q}-\frac{l}{n})-1}\right)$

and

$S_{25}: = \frac{q^{(k-1)(\varepsilon)}}{n \phi(q)}\mathop{\sum\limits_{\chi \mathrm{mod} q\\{\chi \neq \chi_{0}}}} \chi(\overline c)\sum \limits_{l = 1}^{n}\prod\limits_{i = 1}^{k-1}\left(\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G(r_{i}, \chi) \right) \left(\frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e}(\frac{r_{k}}{q}-\frac{l}{n})-1}\right).$

By the same argument of $S_{23}$ , it follows that

$S_{24} \ll q^{\frac{k-1}{2}-\varepsilon}d^{2k}(q)(\log q+n)^{k},$

$S_{25} \ll q^{\frac{k-3}{2}+\varepsilon}(\log q+n).$

Since $n\ll q^{\frac{1}{3}}$ , we have

$\begin{equation} S_{25} \ll S_{24} \ll S_{23} \ll q^{\frac{k-1}{2}+\varepsilon}n^{k}\ll q^{k-2+\varepsilon}. \end{equation}$

(3.9)

Taking $n = 1$ , we get

$\begin{equation} S_{12}\ll q^{\frac{k-1}{2}+\varepsilon}. \end{equation}$

(3.10)

With (3.1), (3.2), (3.9) and (3.10), the proof is complete.

4. Conclusions

This paper considers the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals. And we give an asymptotic formula by the properties of Beatty sequences and the estimates for hyper Kloosterman sums.

Acknowledgment

This work is supported by Natural Science Foundation No. 12271422 of China. The authors would like to express their gratitude to the referee for very helpful and detailed comments.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1]	R. K. Guy, Unsolved problems in number theory, 3 Eds., New York: Springer-Verlag, 2004. https://doi.org/10.1007/978-0-387-26677-0
[2]	W. Zhang, A problem of D. H. Lehmer and its generalization. Ⅱ, Compositio Math., 91 (1994), 47–56.
[3]	W. Zhang, On a problem of D. H. Lehmer and its generalization, Compositio Math., 86 (1993), 307–316.
[4]	W. Zhang, On D. H. Lehmer problem, Chin. Sci. Bull., 37 (1992), 1351–1354.
[5]	Y. Lu, Y. Yi, On the generalization of the D. H. Lehmer problem, Acta Math. Sin.-Engl. Ser., 25 (2009), 1269–1274. https://doi.org/10.1007/s10114-009-7652-3 doi: 10.1007/s10114-009-7652-3
[6]	H. Liu, W. Zhang, Two generalizations of a problem of Lehme, Acta Math. Sin. (Chin. Ser.), 49 (2006), 95–104.
[7]	Z. Guo, Y. Yi, The Lehmer problem and Beatty sequences, submitted for publication, 2022.
[8]	I. Vinogradov, A new estimate of a certain sum containing primes, Rec. Math., 2 (1937), 783–792.
[9]	L. Weinstein, The hyper-Kloosterman sum, Enseign. Math., 27 (1981), 29–40.
[10]	C. Pan, Goldbach conjecture, Beijing: Science Press, 1981.

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AIMS Mathematics

High-dimensional Lehmer problem on Beatty sequences

Related Papers:

Abstract

1. Introduction

2. Preliminary lemmas

3. Proof of theorem

3.1. Estimation of $S_{11}$

3.2. Estimation of $S_{21}$

3.3. Estimation of $S_{22}$ and $S_{12}$

4. Conclusions

Acknowledgment

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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AIMS Mathematics

High-dimensional Lehmer problem on Beatty sequences

Related Papers:

Abstract

1. Introduction

2. Preliminary lemmas

3. Proof of theorem

3.1. Estimation of S11 S_{11}

3.2. Estimation of S21 S_{21}

3.3. Estimation of S22 S_{22} and S12 S_{12}

4. Conclusions

Acknowledgment

Conflict of interest

References

Reader Comments

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3.1. Estimation of $S_{11}$

3.2. Estimation of $S_{21}$

3.3. Estimation of $S_{22}$ and $S_{12}$