# On the Structure of Sets with Few Three-Term Arithmetic Progressions

### Abstract

Fix a prime $p \geq 3$, and a real number $0 < \alpha \leq 1$. Let $S \subset {\mathbb F}_p^n$ be any set with the least number of solutions to $x + y = 2z$ (note that this means that $x,z,y$ is an arithmetic progression), subject to the constraint that $|S| \geq \alpha p^n$. What can one say about the structure of such sets $S$? In this paper we show that they are "essentially" the union of a small number of cosets of some large-dimensional subspace of ${\mathbb F}_p^n$.