On a Furstenberg-Katznelson-Weiss Type Theorem over Finite Fields

Abstract

Using Fourier analysis, Covert, Hart, Iosevich, and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is ≥ ρq 2, with $${q^{-1/2} \ll \rho \leq 1}$$ then it contains an isometric copy of ≥ cρq 3 triangles. In this note, we give a graph theoretic proof of this result.