A note on arithmetic progressions with restricted differences

Abstract

In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd prime power, there is ${\epsilon }_q>0$ such that if $S\subseteq {\mathbb{F}}_q$ with $0\in S$ and $|S|>(q+1)/2$ and $A\subseteq {\mathbb{F}}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $|A|\le q^{(1-{\epsilon }_q)n}$.