Sets avoiding $p$-term arithmetic progressions in ${\mathbb Z}_{q}^n$ are exponentially small

Abstract

Pach and Palincza proved the following generalization of Ellenberg and Gijswijt's bound for the size of $k$-term arithmetic progression-free subsets, where $k\in \{4,5,6\}$: Let $m>0$ be an integer such that $6$ divides $m$ and let $k\in \{4,5,6\}$. Then $$ r_k({\mathbb Z}_{m}^n)\leq (0.948m)^n $$ if $n$ is sufficiently large. Building on the proof technique of Pach and Palincza's upper bound we generalize the Ellenberg and Gijswijt's bound in the following way: Let $p>2$ be any integer and let $q>2$ be a prime. Suppose that $p\leq q$. Then the there exists an $n_0\in \mathbb N$ integer and a $0<\delta(p,q)<1$ real number such that $$ r_p({\mathbb Z}_{q}^n)\leq (\delta(p,q)q)^n $$ for each $n>n_0$.