Erdős-Ginzburg-Ziv constants by avoiding three-term arithmetic progressions

Abstract

For a finite abelian group $G$, the Erdős-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length $\exp (G)$. For a prime $p$, let $r({\mathbb{F}}_p^n)$ denote the size of the largest subset of ${\mathbb{F}}_p^n$ without a three-term arithmetic progression. Although similar methods have been used to study $\mathfrak{s}(G)$ and $r({\mathbb{F}}_p^n)$, no direct connection between these quantities has previously been established. We give an upper bound for $\mathfrak{s}(G)$ in terms of $r({\mathbb{F}}_p^n)$ for the prime divisors $p$ of $\exp (G)$. For the special case $G={\mathbb{F}}_p^n$, we prove $\mathfrak{s}({\mathbb{F}}_p^n)\le 2p\cdot r({\mathbb{F}}_p^n)$. Using the upper bounds for $r({\mathbb{F}}_p^n)$ of Ellenberg and Gijswijt, this result improves the previously best known upper bounds for $\mathfrak{s}({\mathbb{F}}_p^n)$ given by Naslund.

7 pages, minor updates