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

TL;DR

The paper proves that subsets of vector spaces over finite fields avoiding certain full-rank three-point patterns are exponentially small, extending previous results and applying to equilateral triangle avoidance when 3 is a square in the field.

Contribution

It generalizes existing theorems on pattern avoidance in finite field vector spaces to broader classes of patterns and provides explicit exponential bounds.

Findings

Subsets avoiding full-rank three-point patterns are exponentially small.

Sets avoiding equilateral triangles are exponentially small when 3 is a square in the field.

Provides explicit bounds on the size of such subsets.

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 $0.8414 q \leq c_q \leq 0.9184 q$. This generalizes a theorem of Kova\u{c} 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.