Sarkozy's theorem in function fields

TL;DR

This paper extends Sárközy's theorem to polynomials over finite fields, establishing bounds for the existence of polynomial differences that are perfect k-th powers within dense sets.

Contribution

It provides a quantitative version of Sárközy's theorem for polynomials over finite fields with explicit bounds depending polynomially on parameters.

Findings

Dense sets of polynomials contain two with difference a k-th power.

Established bounds depend polynomially on the size of the set and parameters.

The results improve understanding of polynomial differences in finite field settings.

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 $b \in \mathbb{F}_q[T]$.