A Generalization of Sárközy's theorem in function fields

Abstract

Sárközy's theorem says that if $A\subset \mathbb{Z}$ has positive upper asymptotic density, then there are distinct $a_1,a_2\in A$ and $n\in \mathbb{Z}$ such that $a_1-a_2=n^2$. The same is true if $n^2$ is replaced by $F(n)$ for any polynomial $F\in \mathbb{Z}[x]$ with constant term zero. Green proved an ${\mathbb{F}}_q[t]$-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial $F\in {\mathbb{F}}_q[x]$. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in ${\mathbb{F}}_q[t]$, while pointing out that the technical condition can be removed by means of a simple observation.