The zero-sum constant, the Davenport constant and their analogues
Autor
Zakarczemny, Maciej
Opublikowane w
Technical Transactions
Numeracja
Vol. 117, iss. 1
Data wydania
2020
Miejsce wydania
Kraków
Wydawca
Wydawnictwo Politechniki Krakowskiej
Sekcja czasopisma
Mathematics
Język
angielski
eISSN
2353-737X
DOI
https://doi.org/10.37705/TechTrans/e2020027
Słowa kluczowe
zero-sum sequence, Davenport constant, finite Abelian group
Abstrakt
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.
