The zero-sum constant, the Davenport constant and their analogues

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 E_m(G) (or η_m(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 E_m(G) = D(G) – 1 + m|G|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (E_m(G))_m≥1 and (η_m(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.