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

TL;DR

This paper extends bounds on the size of subsets in finite abelian groups avoiding p-term arithmetic progressions, showing these sets are exponentially small in size for large dimensions.

Contribution

It generalizes Ellenberg and Gijswijt's bound to arbitrary primes and integers, providing exponential decay bounds for progression-free sets in ${f Z}_q^n$.

Findings

Sets avoiding p-term progressions are exponentially small in size.

Established bounds depend on parameters p and q.

Results hold for sufficiently large n.

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 $n>n_0$.