Spherical Configurations over Finite Fields

TL;DR

This paper proves that large subsets of finite fields contain all spherical configurations of a certain size, extending geometric combinatorics over finite fields.

Contribution

It establishes conditions under which large subsets of finite fields contain all spherical configurations of a given size, a new result in finite field geometry.

Findings

Large subsets contain all spherical configurations of size k+2

Conditions depend on the dimension and size of the subset

Results hold for sufficiently large odd finite fields

Abstract

We establish that if $d \geq 2k + 6$ and $q$ is odd and sufficiently large with respect to $\alpha \in (0,1)$, then every set $A\subseteq \mathbf{F}_q^d$ of size $|A| \geq \alpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.