Spherical Configurations over Finite Fields

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 $\mathrm{\mid }A\mathrm{\mid }\ge \alpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.