Result: Long sequences having no two nonempty zero-sum subsequences of distinct lengths

Summary: Let \(G\) be an additive finite abelian group. We denote by \(\textbf{disc}(G)\) the smallest positive integer \(t\) such that every sequence \(S\) over \(G\) of length \(|S|\geq t\) has two nonempty zero-sum subsequences of distinct lengths. In this paper, we first extend the list of the groups \(G\) for which \(\textbf{disc}(G)\) is known. Then we focus on the inverse problem associated with \(\textbf{disc}(G)\). Let \(\mathcal{L}_1(G)\) denote the set of all positive integers \(t\) with the property that there is a sequence \(S\) over \(G\) with \(|S|=\textbf{disc}(G)-1\) such that all nonempty zero-sum subsequences of \(S\) have the same length \(t\). We determine \(\mathcal{L}_1(G)\) for some special groups including the groups with large exponents compared to \(|G|/\exp (G)\), the groups of rank at most 2, the groups \(C_{p^n}^r\) with \(3\leq r\leq p\), and the groups \(C_{mp^n}\oplus H\), where \(H\) is a \(p\)-group with \(\mathsf{D}(H)\leq p^n\), and \(\mathsf{D}(H)\) denotes the Davenport constant of \(H\). In particular, we find some groups \(G\) with \(|\mathcal{L}_1(G)|\geq 2\), which disproves a recent conjecture of \textit{W. Gao} et al. [Colloq. Math. 144, No. 1, 31--44 (2016; Zbl 1409.11018)]. Let \(S\) be a sequence over \(G\) such that all nonempty zero-sum subsequences have the same length. We determine the structure of \(S\) for the cyclic group \(C_n\) when \(|S|\geq n+1\), and for the group \(C_n\oplus C_n\) when \(|S|=3n-2=\textbf{disc}(C_n\oplus C_n)-1\).