Sárközy's Theorem in Function Fields

Abstract

Sarkozy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sarkozy's theorem for polynomials over $\mathbb{F}_q$ with polynomial dependencies in the parameters. More precisely, let $P_{q,n}$ be the space of polynomials over $\mathbb{F}_q$ of degree $ 2q^{(1 - c(k,q))n}$, then $A$ contains distinct polynomials $p(T), p'(T)$ such that $p(T) - p'(T) = b(T)^k$ for some $b \in \mathbb{F}_q[T]$.