BRK-type sets over finite fields

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.