A note on arithmetic progressions with restricted differences

TL;DR

This paper extends the Ellenberg--Gijswijt theorem using Tao's slice rank method to analyze subsets of finite fields avoiding arithmetic progressions with restricted differences, providing bounds on their size.

Contribution

It adapts the slice rank method to handle restricted difference sets, extending known results on cap sets to new classes of progression-free subsets.

Findings

If |S| > (q+1)/2, then subsets avoiding progressions with differences in S^n are exponentially small.

The bound |A| ≤ q^{(1-ε_q)n} holds for some ε_q > 0 depending on q.

The method generalizes the approach to a broader class of progression restrictions.

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 $\varepsilon_q>0$ 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 $|A| \leq q^{(1-\varepsilon_q)n}$.