On almost Cap sets in three variables and the multivariable Cap set problem

Abstract

In this note we prove that almost cap sets $A \subset \mathbb{F}_q^n$, i.e., the subsets of $\mathbb{F}_q^n$ that do not contain too many arithmetic progressions of length three, satisfy that $|A| < c_q^n$ for some $c_q < q$. As a corollary we prove a multivariable analogue of Ellenberg-Gijswijt theorem.