A Furstenberg-Katznelson-Weiss type theorem on (d + 1)-point configurations in sets of positive density in finite field geometries

Abstract

We show that if $E\subset {\mathbb{F}}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements, and $|E|\ge \rho q^d$, where $q^{-\frac{1}{2}}\ll \rho \le 1$, then $E$ contains an isometric copy of at least $c{\rho }^{d-1}{q}^{\left(\genfrac{}{}{0ex}{}{d+1}{2}\right)}$ distinct $(d+1)$-point configurations.