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 $$r_k({\mathbb{Z}}_m^n)\le (0.948m{)}^n$$ 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 $p\le q$. Then the there exists an $n_0\in \mathbb{N}$ integer and a $0<\delta (p,q)<1$ real number such that $$r_p({\mathbb{Z}}_q^n)\le (\delta (p,q)q{)}^n$$ for each $n>n_0$.

there is a fatal error in the proof of Lemma 1.9