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

Abstract

Fix a positive prime power $q$, and let ${\mathbb{F}}_q[t]$ be the ring of polynomials over the finite field ${\mathbb{F}}_q$ with $\text{char}({\mathbb{F}}_q)>2$. Suppose $A\subseteq \{f\in {\mathbb{F}}_q[t]:\text{deg }f\le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to Sárközy'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 $O\left(q^{(1-c/\log N)(N+1)}\right)$ due to Lê and Spencer. An important distinction between Green's argument and ours lies in the properties of exponential sums over function fields, which differ in several interesting ways from their number-field counterparts.

37 pages; accepted for publication in Finite Fields and their Applications