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

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 $n, r \in \N$ 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.