Spherical Configurations over Finite Fields

Abstract

We establish that if $d\ge 2k+6$ 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 $|A|\ge \alpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.