Paley Graphs and S\'ark\"ozy's Theorem In Function Fields

TL;DR

This paper extends Sárközy's theorem to function fields by constructing large sets avoiding k-th power differences and explores the independence number of generalized Paley Graphs, advancing understanding of these combinatorial structures.

Contribution

It provides a lower bound for Sárközy's theorem in function fields and generalizes results on Paley Graphs' independence number, adapting combinatorial methods to new settings.

Findings

Constructed large sets of polynomials avoiding k-th power differences.

Established lower bounds for Sárközy's theorem in function fields.

Generalized results on the independence number of Paley Graphs.

Abstract

S\'ark\"ozy's theorem states that dense sets of integers must contain two elements whose difference is a $k^{th}$ power. Following the polynomial method breakthrough of Croot, Lev, and Pach, Green proved a strong quantitative version of this result for $\mathbb{F}_{q}[T]$. In this paper we provide a lower bound for S\'{a}rk\"{o}zy's theorem in function fields by adapting Ruzsa's construction for the analogous problem in $\mathbb{Z}$. We construct a set $A$ of polynomials of degree $<n$ such that $A$ does not contain a $k^{th}$ power difference with $|A|=q^{n-n/2k}$. Additionally, we prove a handful of results concerning the independence number of generalized Paley Graphs, including a generalization of a claim of Ruzsa, which helps with understanding the limit of the method.