Sets Avoiding Full-Rank Three-Point Patterns in $({\mathbb{F}}_q^n{)}^k$ Are Exponentially Small

Abstract

We prove that if a subset of $({\mathbb{F}}_q^n{)}^k$ (with $q$ an odd prime power) avoids a full-rank three-point pattern $\vec{x},\vec{x}+M_1\vec{d},\vec{x}+M_2\vec{d}$ then it is exponentially small, having size at most $3\cdot c_q^{nk}$ where $0.8414q\le c_q\le 0.9184q$. This generalizes a theorem of Kovauc and complements results of Berger, Sah, Sawhney and Tidor. As a consequence, we prove that if $3$ is a square in ${\mathbb{F}}_q$ then subsets of $({\mathbb{F}}_q^n{)}^2$ avoiding equilateral triangles are exponentially small.

7 pages