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arXiv · 2606.12194

Beating Product Constructions for Linear Equations Over Finite Fields

Paul Hametner·Fred Tyrrell
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Abstract

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

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Paul Hametner, Fred Tyrrell. 2026-06-18. Beating Product Constructions for Linear Equations Over Finite Fields. https://arxiv.org/abs/2606.12194

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