A note on arithmetic progressions with restricted differences

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 $\mathrm{\mid }S\mathrm{\mid }>(q+1)\mathrm{/}2$ and $A\subseteq {\mathbb{F}}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $\mathrm{\mid }A\mathrm{\mid }\le q^{(1-{\epsilon }_q)n}$.