On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression

Abstract

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $\mathbb{F}_q^n$ with no three terms in arithmetic progression by $c^n$ with $c \lt q$. For $q=3$, the problem of finding the largest subset of $\mathbb{F}_3^n$ with no three terms in arithmetic progression is called the cap set problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order $n^{-1-\varepsilon} 3^n$.