<abstract xmlns="http://www.w3.org/1999/xhtml">Let D(<italic>G</italic>) be the Davenport constant of a finite Abelian group <italic>G</italic>. For a positive integer <italic>m</italic> (the case <italic>m</italic> = 1, is the classical case) let E<italic>m</italic>(<italic>G</italic>) (or η<italic>m</italic>(<italic>G</italic>)) be the least positive integer <italic>t</italic> such that every sequence of length <italic>t</italic> in <italic>G</italic> contains <italic>m</italic> disjoint zero-sum sequences, each of length |<italic>G</italic>| (or of length ≤ exp(<italic>G</italic>), respectively). In this paper, we prove that if <italic>G</italic> is an Abelian group, then E<italic>m</italic>(<italic>G</italic>) = D(<italic>G</italic>) – 1 + <italic>m</italic>|<italic>G</italic>|, which generalizes Gao’s relation. Moreover, we examine the asymptotic behaviour of the sequences (E<italic>m</italic>(<italic>G</italic>))<italic>m</italic>≥1 and (η<italic>m</italic>(<italic>G</italic>))<italic>m</italic>≥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.</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.

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

Department of Applied Mathematics, Faculty of Computer Science and Telecommunications

Technical Transactions

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

{"article-title":"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...

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

Published Online: Sep 11, 2020

Page range: -

Received: Feb 24, 2020

Accepted: Sep 03, 2020

DOI: https://doi.org/10.37705/TechTrans/e2020027

Keywords
zero-sum sequence, Davenport constant, finite Abelian group

© 2020 Maciej Zakarczemny, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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

Published Online: Sep 11, 2020

Page range: -

Received: Feb 24, 2020

Accepted: Sep 03, 2020

DOI: https://doi.org/10.37705/TechTrans/e2020027

Keywordszero-sum sequence, Davenport constant, finite Abelian group

© 2020 Maciej Zakarczemny, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
zero-sum sequence, Davenport constant, finite Abelian group