Let D(G) be the Davenport constant of a finite Abelian group G. For a positive integer m (the case m = 1, is the classical case) let Em(G) (or ηm(G)) be the least positive integer t such that every sequence of length t in G contains m disjoint zero-sum sequences, each of length |G| (or of length ≤ exp(G), respectively). In this paper, we prove that if G is an Abelian group, then Em(G) = D(G) – 1 + m|G|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (Em(G))m≥1 and (ηm(G))m≥1. We prove a generalization of Kemnitz’s conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the end, we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.
Halter-Koch, F. (1992). A generalization of Davenport’s constant and its arithmetical applications. Colloq. Math., 63(2), 203–210.
Han, D. (2015). The Erdös-Ginzburg-Ziv Theorem for finite nilpotent groups. Archiv Math., 104, 325–322.
Han, D., Zhang, H. (2019). The Erdös-Ginzburg-Ziv Theorem and Noether number for Cm⋉ϕCmn. J. Number Theory, 198, 159–175.
Hamidoune, Y. (1996). On weighted sums in abelian groups. Discrete Math., 162, 127–132.
Harborth, H. (1973). Ein Extremalproblem für Gitterpunkte. J. Reine Angew. Math., 262, 356–360.
Oh, J.S. and Zhong, Q. (2019). On Erdös-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups, to appear in the Israel Journal of Mathematics. Retrieved from https://arxiv.org/abs/1904.13171 (date of access: 2020/04/21).
Olson, J.E. (1969a). A combinatorial problem on finite abelian groups I. J. Number Theory 1, 8–10.
Olson, J.E. (1969b). A combinatorial problem on finite abelian groups II. J. Number Theory 1, 195–199.
Reiher, Ch. (2007). On Kemnitz’s conjecture concerning lattice-points in the plane. The Ramanujan Journal., 13, 333–337.
Rogers, K. (1963). A combinatorial problem in abelian groups. Proc. Cambridge Philos. Soc., 59, 559–562.
Schmid, W.A. (2011). The inverse problem associated to the Davenport constant for C2⊕C2⊕C2n, and applications to the arithmetical characterization of class groups. Electron. J. Combin., 18(1), 1–42.
Sheikh, A. (2017). The Davenport Constant of Finite Abelian Groups (Thesis). London: University of London.