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Inverse zero-sum problems. III. (English) Zbl 1213.11178

Let \(G\) be a finite abelian group. The Davenport constant \(D(G)\) is the smallest integer \(\ell\in\mathbb N\) such that every sequence \(S\) over \(G\) of length \(|S|\geq\ell\) has a nontrivial zero-sum subsequence. Inverse zero-sum problems ask for the structure of sequences that are extremal with respect to a certain property.
In this paper the inverse problem with respect to the Davenport constant and the structure of minimal zero-sum sequences having length \(D(G)\) are studied. Let \(G\) be a group of the form \(G= C_n\oplus C_n\) with \(n\geq 2\). Then \(D(G)= 2n- 1\), and the inverse problem with respect to \(G\) was first studied by W. D. Gao and A. Geroldinger in some articles [see Part I (with W. A. Schmid), Acta Arith. 128, No. 3, 245–279 (2007; Zbl 1188.11052), Part II, W. A. Schmid, Acta Arith. 143, No. 4, 333–343 (2010; Zbl 1219.11151)].
A group \(G\) has the property \(B\) if every minimal zero-sum sequence \(S\) over \(G\) of length \(|S|= 2n- 1\) contains an element with multiplicity \(n- 1\). The main result of this paper is the following
Theorem: Let \(G = C_{mn}\oplus C_{mn}\) with \(m,n\in\mathbb N\) odd. If both \(C_m\oplus JC_m\) and \(C_n\oplus C_n\) have property \(B\), then \(G\) has property \(B\).

MSC:

11P70 Inverse problems of additive number theory, including sumsets
11B50 Sequences (mod \(m\))
11B75 Other combinatorial number theory
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