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 $\mathrm{\ell }$ such that every sequence $S$ over $G$ of length $\mathrm{\ell }$ has a zero-sum subsequence $T$ of length $\mathrm{\mid }T\mathrm{\mid }\in L$. 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.