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

TL;DR

This paper investigates the Erdős–Ginzburg–Ziv constant for high-rank finite abelian groups, especially groups of the form C_n^r, using a novel approach that combines direct and inverse problem techniques.

Contribution

It introduces a new method for studying the Erdős–Ginzburg–Ziv constant in high-rank groups, extending understanding beyond known cases of rank at most two.

Findings

Derived bounds for the Erdős–Ginzburg–Ziv constant of groups C_n^r.

Established connections between direct and inverse problems in zero-sum theory.

Provided insights into the structure of zero-sum sequences in high-rank groups.

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 $n, r \in \N$ 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.