On an analogue of BRK-type sets in finite fields

A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ contains a hypersphere of every radius. In $\mathbb{F}_q^n$, BRK-type sets of degree $\ell$ analogously contain a family of $(n-1)$-dimensional surfaces, parametrized by a dilation factor and determined by a fixed homogeneous polynomial of degree $\ell$. We define $(n,d)$-BRK-type sets of degree $\ell$, which contain a family of $d$-dimensional sets parametrized by an $(n-d)$-dimensional dilation factor and determined by fixed homogeneous polynomials of degree $\ell$. We use the polynomial method to obtain a lower bound $|S| \gtrsim_{n, \ell} q^n$ on $(n,d)$-BRK-type sets $S$ of degree $\ell$. We obtain an improved lower bound $|S| \geq \frac{(q-1)^n}{(\ell + 1 - 2\ell/q)^n}$ by implementing the method of multiplicities; this is the same bound obtained by Trainor on BRK-type sets of degree $\ell$, and we obtain this bound independently of $d$.