An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite field geometries

Abstract

We prove that if the cardinality of a subset of the 2-dimensional vector space over a finite field with $q$ elements is $\ge ρq^2$, with $\frac{1}{\sqrt{q}}<<ρ\leq 1$, then it contains an isometric copy of $\ge cρq^3$ triangles.