On an analogue of BRK-type sets in finite fields
Madeline Forbes
TL;DR
This work develops a finite-field analogue of Besicovitch-Rado-Kinney (BRK) sets by defining $(n,d)$-BRK-type sets of degree $\ell$ in $\mathbb{F}_q^n$, which contain a family of $d$-dimensional sets governed by homogeneous polynomials. It employs the polynomial method and the method of multiplicities to derive two main lower bounds on the size of such sets: a baseline bound $|S| \gtrsim_{n,\ell} q^n$ and an improved multiplicity-based bound $|S| \geq \frac{(q-1)^n}{(\ell + 1 - 2\ell/q)^n}$, the latter matching Trainor's earlier results and independent of $d$. The authors adapt vanishing-multiplicity techniques to the $(n,d)$-BRK-type context and utilize a graded lexicographic order to control leading terms, enabling coefficient extraction and contradiction arguments. Collectively, the results advance the finite-field BRK-type theory by providing robust, $d$-independent lower bounds and connecting BRK-type phenomena to the polynomial method and multiplicities in higher-dimensional families.
Abstract
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 $d$.