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

الأصل · EN

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