Dot-product graphs in finite fields

TL;DR

This paper investigates the properties of dot-product graphs over finite fields, demonstrating that large enough product sets lead to graphs with positive density using Fourier analysis techniques.

Contribution

It introduces finite field Fourier analytic methods to analyze dot-product graphs and establishes conditions for positive density based on product set size.

Findings

Large product sets imply positive density in dot-product graphs.

Fourier analysis effectively studies combinatorial properties in finite fields.

Provides new tools for understanding geometric structures in finite fields.

Abstract

In this paper, we study the dot-product graphs in $\mathbb{F}_q^d$. We prove that if the size of the product of two adjacent sets is large enough, then the set of dot-product graphs has positive density. Our method is based on finite field Fourier analytic techniques.