Soft order topology and graph comparison based on soft order

Kemal Taşköprü; Kemal Taşköprü

doi:10.3934/math.2023492

AIMS Mathematics

2023, Volume 8, Issue 4: 9761-9781. doi: 10.3934/math.2023492

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Research article

Soft order topology and graph comparison based on soft order

Kemal Taşköprü ^,

Department of Mathematics, Faculty of Science, Bilecik Şeyh Edebali University, Bilecik 11100, Turkey

Received: 11 December 2022 Revised: 07 February 2023 Accepted: 15 February 2023 Published: 21 February 2023
MSC : 03E72, 06F30, 05C90

Soft sets provide a suitable framework for representing and dealing with vagueness. A scenario for vagueness can be that alternatives are composed of specific factors and these factors have specific attributes. Towards this scenario, this paper introduces soft order and its associated order topology on the soft sets with a novel approach. We first present the definitions and properties of the soft order relations on the soft sets via soft elements. Next, we define soft order topology on any soft set and provide some properties of this topology. In order to implement what we introduced about the soft orders, we describe soft preference and soft utility mapping on the soft sets and we finally demonstrate a decision-making application over the soft orders intended for comparing graphs.

Keywords:

Citation: Kemal Taşköprü. Soft order topology and graph comparison based on soft order[J]. AIMS Mathematics, 2023, 8(4): 9761-9781. doi: 10.3934/math.2023492

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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.

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3.1. Estimation of $S_{11}$

3.2. Estimation of $S_{21}$

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3.1. Estimation of $S_{11}$

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