S\'ark\"ozy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property

TL;DR

This paper extends Sárközy's theorem to the polynomial ring over finite fields, establishing bounds on sets avoiding differences of the form P-1 with P irreducible, using the van der Corput property.

Contribution

It adapts Green's approach to Sárközy's theorem in the setting of finite fields, providing improved bounds on the size of such sets.

Findings

Established a bound |A| ≪ q^{(N+1)(11/12+o(1))} for sets avoiding differences of the form P-1.

Improved upon previous bounds by Lê and Spencer for the size of these sets.

Extended Sárközy's theorem techniques to polynomial rings over finite fields.

Abstract

Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]\colon\text{deg}~ f \le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to S\'ark\"ozy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[|A| \ll q^{(N+1)(11/12+o(1))},\] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to L\^{e} and Spencer.