S\'ark\"ozy's Theorem in Various Finite Field Settings

TL;DR

This paper extends Green's polynomial ring analogue of Sarkozy's theorem by removing root-count restrictions and applies the results to subsets of finite fields, providing stronger bounds in these algebraic settings.

Contribution

It generalizes Green's result on Sarkozy's theorem in polynomial rings over finite fields by removing the root count condition, and applies it to subsets of finite fields for large extension degrees.

Findings

Stronger bounds for subsets avoiding polynomial differences in $\\mathbb{F}_q[x]$

Removal of the coprimality condition on roots in Green's theorem

Application to subsets of finite fields with large extension degree

Abstract

In this paper, we strengthen a result by Green about an analogue of Sarkozy's theorem in the setting of polynomial rings $\mathbb{F}_q[x]$. In the integer setting, for a given polynomial $F \in \mathbb{Z}[x]$ with constant term zero, (a generalization of) Sarkozy's theorem gives an upper bound on the maximum size of a subset $A \subset \{1, \ldots, n \}$ that does not contain distinct $a_1,a_2 \in A$ satisfying $a_1 - a_2 = F(b)$ for some $ b \in \mathbb{Z}$. Green proved an analogous result with much stronger bounds in the setting of subsets $A \subset \mathbb{F}_q[x]$ of the polynomial ring $\mathbb{F}_q[x]$, but required the additional condition that the number of roots of the polynomial $F \in \mathbb{F}_q[x]$ is coprime to $q$. We generalize Green's result, removing this condition. As an application, we also obtain a version of Sarkozy's theorem with similarly strong bounds for subsets $A \subset \mathbb{F}_q$ for $q = p^n$ for a fixed prime $p$ and large $n$.