On the Energy of Roots

IF 0.6 4区数学 Q3 MATHEMATICS Mathematical Notes Pub Date : 2024-07-15 DOI:10.1134/s0001434624050250

A. S. Volostnov

引用次数: 0

Abstract

An estimate of the additive energy of roots modulo a prime for sets with small doubling that has recently been obtained by Zaharescu, Kerr, Shkredov, and Shparlinskii is improved. The problem of determining the maximum cardinalities of the sets $|A+A|$ and $|f(A)+f(A)|$, where $f$ is a polynomial of small degree and $A$ is a subset of a finite field whose size is sufficiently small in comparison with the characteristic of the field, is also considered. In particular, it is proved that

$$\max(|A+A|,|A^3+A^3|)\ge|A|^{16/15},$$

$\max(|A+A|,|A^4+A^4|)\ge|A|^{25/24}$, and $\max(|A+A|,|A^5+A^5|)\ge|A|^{25/24}$.

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根的能量

摘要扎哈里斯库（Zaharescu）、克尔（Kerr）、什克雷多夫（Shkredov）和什帕林斯基（Shparlinskii）最近得到的关于具有小倍增的集合的根模的加法能量的估计值得到了改进。还考虑了确定集合 $|A+A|$ 和 $|f(A)+f(A)|$的最大心数的问题，其中 $f$ 是小度多项式，$A$ 是有限域的子集，其大小与域的特征相比足够小。特别是，证明了 $$\max(|A+A|,|A^3+A^3|)ge|A|^{16/15}，$$\max(|A+A|,|A^4+A^4|)\ge|A|^{25/24}\)，以及$\max(|A+A|,|A^5+A^5|)\ge|A|^{25/24}$。

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

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.