Affiliation:
1. Department of Applied Mathematics, Faculty of Computer Science and Telecommunications, Cracow University of Technology
Abstract
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.
Publisher
Cracow University of Technology
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