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arXiv · 2403.19824

BRK-type sets over finite fields

Charlotte Trainor
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Abstract

A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ 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, $\mathbb{F}_q$. We define BRK-type sets in $\mathbb{F}_q^n$, 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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Charlotte Trainor. 2024-03-28. BRK-type sets over finite fields. https://arxiv.org/abs/2403.19824

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