On a Furstenberg-Katznelson-Weiss type theorem over finite fields

TL;DR

This paper provides a graph theoretic proof for a finite field geometric theorem that guarantees the presence of many triangles in large enough subsets, extending Fourier analysis results with a combinatorial approach.

Contribution

It introduces a graph theoretic proof for a known Fourier analysis result on triangle configurations in finite fields.

Findings

Proves the finite field triangle theorem using graph theory.

Establishes the existence of many triangles in large subsets.

Provides an alternative proof method to Fourier analysis.

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 >= rq^2, with q^{-1/2} << r <= 1 then it contains an isometric copy of >= crq^3 triangles. In this note, we give a graph theoretic proof of this result.