  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.
