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 $\mathrm{\ell }$ analogously contain a family of $(n-1)$-dimensional surfaces, parametrized by a dilation factor and determined by a fixed homogeneous polynomial of degree $\mathrm{\ell }$. We define $(n,d)$-BRK-type sets of degree $\mathrm{\ell }$, which contain a family of $d$-dimensional sets parametrized by an $(n-d)$-dimensional dilation factor and determined by fixed homogeneous polynomials of degree $\mathrm{\ell }$. We use the polynomial method to obtain a lower bound $\mathrm{\mid }S\mathrm{\mid }{\gtrsim }_{n,\mathrm{\ell }}{q}^n$ on $(n,d)$-BRK-type sets $S$ of degree $\mathrm{\ell }$. We obtain an improved lower bound $\mathrm{\mid }S\mathrm{\mid }\ge \frac{(q-1{)}^n}{(\mathrm{\ell }+1-2\mathrm{\ell }\mathrm{/}q{)}^n}$ by implementing the method of multiplicities; this is the same bound obtained by Trainor on BRK-type sets of degree $\mathrm{\ell }$, and we obtain this bound independently of $d$.