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 ≤ e...
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| Format: | Article |
| Language: | English |
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Sciendo
2020-01-01
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| Series: | Technical Transactions |
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| Online Access: | https://doi.org/10.37705/TechTrans/e2020027 |
| _version_ | 1797387190593912832 |
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| author | Zakarczemny Maciej |
| author_facet | Zakarczemny Maciej |
| author_sort | Zakarczemny Maciej |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T22:21:29Z |
| format | Article |
| id | doaj.art-67327b8a970f4caa978b791e77b9ec77 |
| institution | Directory Open Access Journal |
| issn | 2353-737X |
| language | English |
| last_indexed | 2024-03-08T22:21:29Z |
| publishDate | 2020-01-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Technical Transactions |
| spelling | doaj.art-67327b8a970f4caa978b791e77b9ec772023-12-18T12:46:11ZengSciendoTechnical Transactions2353-737X2020-01-01117110.37705/TechTrans/e2020027The zero-sum constant, the Davenport constant and their analoguesZakarczemny Maciej0Department of Applied Mathematics, Faculty of Computer Science and Telecommunications, Cracow University of TechnologyLet 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.https://doi.org/10.37705/TechTrans/e2020027zero-sum sequencedavenport constantfinite abelian group |
| spellingShingle | Zakarczemny Maciej The zero-sum constant, the Davenport constant and their analogues Technical Transactions zero-sum sequence davenport constant finite abelian group |
| title | The zero-sum constant, the Davenport constant and their analogues |
| title_full | The zero-sum constant, the Davenport constant and their analogues |
| title_fullStr | The zero-sum constant, the Davenport constant and their analogues |
| title_full_unstemmed | The zero-sum constant, the Davenport constant and their analogues |
| title_short | The zero-sum constant, the Davenport constant and their analogues |
| title_sort | zero sum constant the davenport constant and their analogues |
| topic | zero-sum sequence davenport constant finite abelian group |
| url | https://doi.org/10.37705/TechTrans/e2020027 |
| work_keys_str_mv | AT zakarczemnymaciej thezerosumconstantthedavenportconstantandtheiranalogues AT zakarczemnymaciej zerosumconstantthedavenportconstantandtheiranalogues |