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 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.

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Section "Mathematics"

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Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).