On the Erd{\H o}s--Ginzburg--Ziv constant of finite abelian groups of high rank

Abstract

Let $G$ be a finite abelian group. The Erd{\H o}s--Ginzburg--Ziv constant $\mathsf{s}(G)$ of $G$ is defined as the smallest integer $l\in \mathbb{N}$ such that every sequence \ $S$ \ over $G$ of length $|S|\ge l$ \ has a zero-sum subsequence $T$ of length $|T|=\exp (G)$. If $G$ has rank at most two, then the precise value of $\mathsf{s}(G)$ is known (for cyclic groups this is the Theorem of Erd{\H o}s-Ginzburg-Ziv). Only very little is known for groups of higher rank. In the present paper, we focus on groups of the form $G=C_n^r$, with $n,r\in \mathbb{N}$ and $n\ge 2$, and we tackle the study of $\mathsf{s}(G)$ with a new approach, combining the direct problem with the associated inverse problem.

10 pages