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

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 << r ￡ 1{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.