Abstract

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 $A\subseteq{\bf F}_q^d$of size $|A|\geq\alpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.