On some congruences and exponential sums

IF 1.2 3区数学 Q1 MATHEMATICS Finite Fields and Their Applications Pub Date : 2024-06-27 DOI:10.1016/j.ffa.2024.102451

Moubariz Z. Garaev , Igor E. Shparlinski

引用次数: 0

Abstract

Let $ε > 0$ be a fixed small constant, $F_{p}$ be the finite field of p elements for prime p. We consider additive and multiplicative problems in $F_{p}$ that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let $M$ be an arbitrary subset of $F_{p}$ . If $# M > p^{1 / 3 + ε}$ and $H ⩾ p^{2 / 3}$ or if $# M > p^{3 / 5 + ε}$ and $H ⩾ p^{3 / 5 + ε}$ then all, but $O (p^{1 - δ})$ elements of $F_{p}$ can be represented in the form hm with $h \in [1, H]$ and $m \in M$ , where $δ > 0$ depends only on ε. Furthermore, let $X$ be an arbitrary interval of length H and s be a fixed positive integer. If $H > p^{17 / 35 + ε}, # M > p^{17 / 35 + ε},$ then the number $T_{6} (λ)$ of solutions to the congruence $\frac{m_{1}}{x_{1}^{s}} + \frac{m_{2}}{x_{2}^{s}} + \frac{m_{3}}{x_{3}^{s}} + \frac{m_{4}}{x_{4}^{s}} + \frac{m_{5}}{x_{5}^{s}} + \frac{m_{6}}{x_{6}^{s}} \equiv λ \mod p, m_{i} \in M, x_{i} \in X, i = 1, \dots, 6,$ satisfies $T_{6} (λ) = \frac{H^{6} {(# M)}^{6}}{p} (1 + O (p^{- δ})) .$

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关于一些全等和指数和

让 ε>0 是一个固定的小常数，Fp 是素数 p 的 p 元素有限域。我们考虑 Fp 中涉及区间和任意集合的加法和乘法问题。我们的代表性结果举例如下。设 M 是 Fp 的任意子集。如果 #M>p1/3+ε 和 H⩾p2/3，或者如果 #M>p3/5+ε 和 H⩾p3/5+ε，那么除了 O（p1-δ）个元素外，Fp 的所有元素都可以用 hm 的形式表示，其中 h∈[1,H]，m∈M，δ>0 只取决于 ε。此外，设 X 是长度为 H 的任意区间，s 是一个固定的正整数。若 H>p17/35+ε,#M>p17/35+ε，则全等m1x1s+m2x2s+m3x3s+m4x4s+m5x5s+m6x6s≡λmodp,mi∈M,xi∈X,i=1,...,6 的解的个数 T6(λ)满足T6(λ)=H6(#M)6p(1+O(p-δ))。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.

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