Inverse zero-sum problem of finite abelian groups of rank 2 (Q6545061)

Let \(G\) be an additive finite abelian group. Let \(\mathcal F(G)\) be the free abelian monoid with basis \(G\). The elements of \(\mathcal F(G)\) are called sequences over \(G\), so that sequences are finite and unordered, the terms belong to \(G\) and repetition is allowed. In this way, sequences can be written as \N\[\NS=g_1\cdot \ldots \cdot g_{\ell} = \prod_{g \in G} g^{\mathsf{v}_g(S)}.\N\]\NHere, \(|S| = \ell\) is the length of \(S\) and \(\mathsf{v}_g(S)\) is the multiplicity of \(g\) in \(S\). Let \(\sigma(S) = g_1+\ldots+g_{\ell} = \sum_{g \in G} \mathsf{v}_g(S)g\) denote the sum of \(S\). We say that \(S\) is a zero-sum sequence if \(\sigma(S) = 0\). A subsequence \(T\) of \(S\) is a divisor of \(S\) in \(\mathcal F(G)\), that is, \(\mathsf{v}_g(T) \le \mathsf{v}_g(S)\) for every \(g \in G\). Let \(\Sigma_k(S) = \{\sigma(T); \; T \text{ is a subsequence of } S \text{ and }|T|=k\}\) denote the set of \(k\)-sums of \(S\).\N\NLet \(\mathsf{D}(G)\) denote the Davenport constant of \(G\), that is, the smallest positive integer \(\ell\) such that every sequence \(S\) of length \(|S| \ge \ell\) over \(G\) has a nonempty zero-sum subsequence.\N\NIn 1961, \textit{P. Erdős} et al. [Bull. Res. Council Israel 10F, 41--43 (1961; Zbl 0063.00009)] proved that if \(S\) is a sequence over \(G\) with \(|S| \ge 2|G|-1\), then \(0 \in \Sigma_{|G|}(S)\).\N\NSimilarly, let \(\mathsf{s}_{|G|}(G)\) denote the smallest integer \(d\) such that every sequence over \(G\) of length \(|S| \ge d\) has a zero-sum subsequence of length \(|G|\). In 1996, \textit{W. D. Gao} [J. Number Theory 58, No. 1, 100--103 (1996; Zbl 0892.11005)] proved that \(\mathsf{s}_{|G|}(G) = |G| + \mathsf{D}(G) - 1\).\N\NSuppose now that \(S\) is a sequence over \(G\) of length \(|S| = |G|+r\) and \(0 \not\in \Sigma_{|G|}(S)\), where \(r \in [0,\mathsf{D}(G)-2]\). A natural problem is to determine a lower bound for \(|\Sigma_{|G|}(S)|\). In this paper, the authors consider finite abelian groups \(G\) of rank \(2\), investigate lower bounds for \(|\Sigma_{|G|}(S)|\), where \(r \in \{\mathsf{D}(G)-3,\mathsf{D}(G)-2\}\) and completely determine the structure of \(S\) when \(|\Sigma_{|G|}(S)|\) is minimal. The main results are the following.\N\NTheorem 1.1. Let \(m,n\) be positive integers with \(n \ge 3\), and \(S\) be a sequence over \(G = C_n \oplus C_{mn}\) with \(|S| = n^2m+mn+n-3\). Suppose that \(0 \not\in \Sigma_{n^2m}(S)\). Then \(|\Sigma_{n^2m}(S)| = n^2m-1\). Moreover, all possible forms for the structure of \(S\) are provided.\N\NThe conclusion of Theorem 1.1 also holds for \(G = C_2 \oplus C_{2m}\), \(m \ge 1\). The proof is similar but the authors omit it.\N\NTheorem 1.3. Let \(m,n\) be positive integers with \(n \ge 3\), and \(S\) be a sequence over \(G = C_n \oplus C_{mn}\) with \(|S| = n^2m+mn+n-4\). Suppose that \(0 \not\in \Sigma_{n^2m}(S)\). Then \(|\Sigma_{n^2m}(S)| \ge n^2m-nm-1\). Furthermore, the authors provide the structure of \(S\) for which equality holds.\N\NThe conclusion of Theorem 1.3 also holds for \(G = C_2 \oplus C_{2m}\), \(m \ge 2\). However, it does not hold when \(G = C_2 \oplus C_2\). The proof is similar but the authors omit it.