Beating Product Constructions for Linear Equations Over Finite Fields

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.

10 pages