Paley graphs and sárközy’s theorem in function fields

Abstract

Abstract Sárközy’s theorem states that dense sets of integers must contain two elements whose difference is a kth power. Following the polynomial method breakthrough of Croot, Lev and Pach [3], Green proved a strong quantitative version of this result for $\mathbb{F}_{q}[T]$. In this paper we provide a lower bound for Sárközy’s theorem in function fields by adapting Ruzsa’s construction [17] for the analogous problem in $\mathbb{Z}$. We construct a set A of polynomials of degree < n such that A does not contain a kth 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.