Fixed point theory in complex valued controlled metric spaces with an application

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:

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

where

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

and obtained an interesting asymptotic formula,

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

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

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

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

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

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:

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

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

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:

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

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

and

Taking

we have

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

Lemma 2.3. Let

Then

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

then

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

3.   Proof of theorem

We begin by the definition

where

and

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

where

and

For $S_{2}$, it follows that

where

and

3.1.   Estimation of $S_{11}$

From the classical bound

and Lemma 2.2, we have

3.2.   Estimation of $S_{21}$

From Lemma 2.2, we obtain

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

By the properties of exponential sums,

Let

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

and

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

Therefore,

where

and

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

Let

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

By Mobius inversion, we get

and

Hence,

From (3.8) we may deduce the following result:

It is easy to see

and we obtain

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

Let

and

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

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

Taking $n = 1$, we get

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.