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

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.