A Furstenberg–Katznelson–Weiss type theorem on -point configurations in sets of positive density in finite field geometries

We show that if E ⊂ F q d , the d -dimensional vector space over the finite field with q elements, and | E | ≥ ρ q d , where q − 1 2 ≪ ρ ≤ 1 , then E contains an isometric copy of at least c ρ d − 1 q d + 1 2 distinct ( d + 1 ) -point configurations.