Sets of Cross Numbers of Sequences over Finite Abelian Groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909134649032704 |
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| author | Bashir, Aqsa Schmid, Wolfgang A. |
| author_facet | Bashir, Aqsa Schmid, Wolfgang A. |
| contents | Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial zero-sum free sequences over $G$. It is clear that $\mathsf W(G)$ and $\mathsf w(G)$ are bounded subsets of $\frac{1}{\exp(G)}\mathbb{N}$ with maximum $ \mathsf K(G)$ and $\mathsf k(G)$, respectively (here $\mathsf{K}(G)$ and $\mathsf{k}(G)$ denote the large and the small cross number of $G$, respectively). We give results on the structure of $\mathsf W(G)$ and $\mathsf w(G)$. We first show that both sets contain long arithmetic progressions and that only close to the maximum there might be some gaps. Then, we provide groups for which $\mathsf W(G)$ and $\mathsf w(G)$ actually are arithmetic progressions, and argue that this is rather a rare phenomenon. Finally, we provide some results in case there are gaps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07138 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sets of Cross Numbers of Sequences over Finite Abelian Groups Bashir, Aqsa Schmid, Wolfgang A. Combinatorics Number Theory 11B30 Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial zero-sum free sequences over $G$. It is clear that $\mathsf W(G)$ and $\mathsf w(G)$ are bounded subsets of $\frac{1}{\exp(G)}\mathbb{N}$ with maximum $ \mathsf K(G)$ and $\mathsf k(G)$, respectively (here $\mathsf{K}(G)$ and $\mathsf{k}(G)$ denote the large and the small cross number of $G$, respectively). We give results on the structure of $\mathsf W(G)$ and $\mathsf w(G)$. We first show that both sets contain long arithmetic progressions and that only close to the maximum there might be some gaps. Then, we provide groups for which $\mathsf W(G)$ and $\mathsf w(G)$ actually are arithmetic progressions, and argue that this is rather a rare phenomenon. Finally, we provide some results in case there are gaps. |
| title | Sets of Cross Numbers of Sequences over Finite Abelian Groups |
| topic | Combinatorics Number Theory 11B30 |
| url | https://arxiv.org/abs/2403.07138 |