The Davenport constant of a box. (English) Zbl 1384.11042
Let \(G\) be an additive abelian group and \(X \subseteq G\) a subset. The Davenport constant \(\mathsf D (X)\) of \(X\) is the supremum over all \(\ell \in \mathbb N\) for which there is a minimal zero-sum sequence over \(X\) having length \(\ell\). If \(X=G\) is finite, then \(\mathsf D (G)\) is a central and well-studied invariant in zero-sum theory. If \(X\) is an arbitrary finite subset of an abelian group, then it is easy to see that \(\mathsf D (X)\) is finite too but so far there were no further results in this direction. Recent applications for direct-sum decompositions of modules raised the need for more precise results on \(\mathsf D (X)\) in case of well-structured subsets \(X\) of finitely generated abelian groups, see [N. R. Baeth and A. Geroldinger, Pac. J. Math. 271, No. 2, 257–319 (2014; Zbl 1347.16006)], and [N. R. Baeth et al., J. Algebra Appl. 14, No. 2, 1550016, 60 p. (2015; Zbl 1385.16003)].
The focus of the present paper is on boxes \(X = [-m_1, m_1] \times \ldots \times [-m_d, m_d] \subset \mathbb Z^d = G\). The authors derive lower and upper bounds for \(\mathsf D (X)\) and study the asymptotic behavior of \(\mathsf D (X)\) when the box is growing.
The focus of the present paper is on boxes \(X = [-m_1, m_1] \times \ldots \times [-m_d, m_d] \subset \mathbb Z^d = G\). The authors derive lower and upper bounds for \(\mathsf D (X)\) and study the asymptotic behavior of \(\mathsf D (X)\) when the box is growing.
Reviewer: Qinghai Zhong (Graz)
MSC:
| 11B75 | Other combinatorial number theory |
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 11P70 | Inverse problems of additive number theory, including sumsets |
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