On some zero-sum invariants for abelian groups of rank three

TL;DR

This paper investigates zero-sum invariants in finite abelian groups of rank three, providing exact values and bounds for specific invariants, and improving upon previous results in the field.

Contribution

It offers new precise calculations and bounds for zero-sum invariants in rank three abelian groups, advancing the understanding of these invariants.

Findings

Derived exact values for certain zero-sum invariants

Established upper bounds for invariants in specific groups

Improved previous results by Gao-Thangadurai and Han-Zhang

Abstract

Let $G$ be an additive finite abelian group with exponent $\exp(G)$. For $L\subseteq \mathbb N$, let $\mathsf{s}_{L}(G)$ be the smallest integer $\ell$ such that every sequence $S$ over $G$ of length $\ell$ has a zero-sum subsequence $T$ of length $|T|\in L$. In this paper, we consider the invariants $\mathsf{s}_{[1,t]}(G)$ and $\mathsf{s}_{\{k\exp(G)\}}(G)$ (with $k\in \mathbb N$). We obtain precise values as well as upper bounds of the above invariants for some abelian groups of rank three. Some of these results improve previous results of Gao-Thangadurai and Han-Zhang.