Sarkozy's theorem in function fields

Abstract

S\'ark\"ozy 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 S\'ark\"ozy'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 $< n$ in an indeterminate $T$. Let $k \geq 2$ be an integer and let $q$ be a prime power. Set $c(k,q) := (2 k^2 D_q(k)^2\log q)^{-1}$, where $D_q(k)$ is the sum of the digits of $k$ in base $q$. If $A \subset P_{q,n}$ is a set with $|A| > 2q^{(1 - c(k,q))n}$, then $A$ contains distinct polynomials $p(T), p'(T)$ such that $p(T) - p'(T) = b(T)^k$ for some $b \in \mathbb{F}_q[T]$.