BRK-type sets over finite fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-03-25 DOI:10.1016/j.ffa.2025.102624

Charlotte Trainor

引用次数: 0

Abstract

A Besicovitch-Rado-Kinney (BRK) set in

R^{n}

is a Borel set that contains a

(n - 1)

-dimensional sphere of radius r, for each

r > 0

. It is known that such sets have Hausdorff dimension n from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field,

F_{q}

. We define BRK-type sets in

F_{q}^{n}

, and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture.

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有限域上的brk型集合

Rn中的Besicovitch-Rado-Kinney （BRK）集合是包含半径为r的（n−1）维球体的Borel集合，对于每个r>；0。从Kolasa和Wolff的工作中我们知道这样的集合具有Hausdorff维数n。本文考虑有限域Fq上的一个类似问题。我们在Fqn中定义了brk型集合，并利用Dvir有限域Kakeya猜想的证明所引入的技术建立了brk型集合大小的下界。

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.