Paley Graphs and Sárközy's Theorem In Function Fields

Abstract

Sárközy'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 Sárközy'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.

7 pages

References in corpus (2)

• On the clique number of Paley graphs of prime power order
• Difference sets and positive exponential sums II: cubic residues in cyclic groups

Cited by in corpus (1)

• Intersective sets over abelian groups