Exponential Bounds for the Erd\H{o}s-Ginzburg-Ziv Constant

TL;DR

This paper establishes new exponential upper bounds for the Erd ext{"o}s-Ginzburg-Ziv constant of finite abelian groups using partition rank techniques, improving previous bounds for large groups and primes.

Contribution

It introduces a novel application of partition rank to bound the Erd ext{"o}s-Ginzburg-Ziv constant, providing the first exponential improvement for large groups and primes.

Findings

New exponential bounds for $rak{s}(bF_p^n)$ for odd primes p.

Conditional near-optimal bounds assuming property D.

First exponential improvement over trivial bounds for large n and p>3.

Abstract

The Erd\H{o}s-Ginzburg-Ziv constant of an abelian group $G$, denoted $\mathfrak{s}(G)$, is the smallest $k\in\mathbb{N}$ such that any sequence of elements of $G$ of length $k$ contains a zero-sum subsequence of length $\exp(G)$. In this paper, we use the partition rank, which generalizes the slice rank, to prove that for any odd prime $p$, \[ \mathfrak{s}\left(\mathbb{F}_{p}^{n}\right)\leq(p-1)2^{p}\left(J(p)\cdot p\right)^{n} \] where $0.8414<J(p)<0.91837$ is the constant appearing in Ellenberg and Gijswijt's bound on arithmetic progression-free subsets of $\mathbb{F}_{p}^{n}$. For large $n$, and $p>3$, this is the first exponential improvement to the trivial bound. We also provide a near optimal result conditional on the conjecture that $\left(\mathbb{Z}/k\mathbb{Z}\right)^{n}$ satisfies property $D$, showing that in this case \[ \mathfrak{s}\left(\left(\mathbb{Z}/k\mathbb{Z}\right)^{n}\right)\leq(k-1)4^{n}+k. \]