Result: On tiny zero-sum sequences over finite abelian groups

Let \((G,+,0)\) be a finite abelian group. A sequence over \(G\) is a finite unordered sequence with terms from \(G\) and repetition allowed. Let \[ S=g_1\cdot g_2\cdot\ldots\cdot g_{\ell} \] be a sequence over \(G\) of length \(\ell\). If \(\ell\ge 1\), \(\sigma(S):=g_1+\ldots+g_{\ell}=0\), and \(\mathsf k(S):=\frac{1}{\operatorname{ord}(g_1)}+\ldots+\frac{1}{\operatorname{ord}(g_{\ell})}\le 1\), then we say \(S\) is a tiny zero-sum sequence. We denote by \(\mathsf t(G)\) the smallest integer \(t\) such that every sequence of length \(t\) has a tiny zero-sum subsequence. The investigations on \(\mathsf {t}(G)\) originate in a conjecture, addressed by Erdős and Lemke in the late 1980s. Among others, we know that \begin{itemize} \item if \(G\) is cyclic, then \(\mathsf t(G)=|G|\); \item if \(G\cong C_{p^{\alpha}}^2\), where \(p\) is a prime and \(\alpha\) is a positive integer, then \(\mathsf t(G)=3p^{\alpha}-2\). \end{itemize} \textit{B. Girard} [Adv. Math. 231, No. 3--4, 1843--1857 (2012; Zbl 1251.05178)] conjectured that \(\mathsf t(G)=2m+mn-2\) for all groups \(G\cong C_m\oplus C_{mn}\), where \(m,n\) are positive integers. In this manuscript, the authors confirmed this conjecture for groups \(G\cong C_m\oplus C_m\) such that \(m=p_1^{\alpha_1}\cdot\ldots\cdot p_{s}^{\alpha_s}\) with \(\frac{1}{p_1}+\ldots+\frac{1}{p_{s}}< 1\), where \(p_1,\ldots, p_s\) are distinct primes and \(s, \alpha_1, \ldots, \alpha_s\) are positive integers (see Theorem 1.4).