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

We show that if E⊂Fqd, the dd-dimensional vector space over the finite field with qq elements, and |E|≥ρqd|E|≥ρqd, where q−12≪ρ≤1, then EE contains an isometric copy of at least cρd−1qd+12 distinct (d+1)(d+1)-point configurations.