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arXiv · 2510.27581

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

Steve Fan·Andrew Lott
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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 $A \subseteq \{f \in \mathbb{F}_q[t]: \text{deg } f \leq N\}$ 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. 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.

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Steve Fan, Andrew Lott. 2025-10-31. S\'ark\"ozy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property. https://arxiv.org/abs/2510.27581

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