Sárközy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property
Steve Fan, Andrew Lott
TL;DR
This work analyzes Sárközy-type sets in the function field setting by transferring Green’s shifted-primes van der Corput framework to \\mathbb{F}_q[t]. It develops a function-field circle method with major/minor arcs and Ramanujan-type expansions to control correlations of shifted irreducibles, proving the upper bound \\lvert A \ vert \\ll q^{(N+1)(11/12+o(1))} and a matching lower bound via a Ruzsa-type construction. The approach hinges on constructing a van der Corput proxy T, establishing Fourier closeness between proxies for primes, and performing a triple-correlation analysis of Ramanujan expansions to manage major-arc contributions. The results substantially improve prior quasi-polynomial bounds in the q-aspect and demonstrate the van der Corput property for shifted irreducibles in function fields, with potential implications for related Sárközy-type problems in other global 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 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 $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to Lê and Spencer.