The structure of sequences with zero-sum subsequences of the same length on finite abelian groups of rank two

Abstract

Let $G$ be an additive finite abelian group, and let $\mathrm{disc}(G)$ denote the smallest positive integer $t$ with the property that every sequence $S$ over $G$ with length $|S|\ge t$ contains two nonempty zero-sum subsequences of distinct lengths. In recent years, Gao et al. established the exact value of $\mathrm{disc}(G)$ for all finite abelian groups of rank $2$ and resolved the corresponding inverse problem for the group $C_n\oplus C_n$. In this paper, we characterize the structure of sequences $S$ over $G=C_n\oplus C_{nm}$ (where $m\ge 2$) when $|S|=\mathrm{disc}(G)-1$ and all nonempty zero-sum subsequences of $S$ have the same length.