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Preprints, Working Papers, ... Year : 2024

SETS OF CROSS NUMBERS OF SEQUENCES OVER FINITE ABELIAN GROUPS

Aqsa Bashir
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Abstract

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.
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Dates and versions

hal-04546837 , version 1 (15-04-2024)

Identifiers

  • HAL Id : hal-04546837 , version 1

Cite

Aqsa Bashir, Wolfgang Schmid. SETS OF CROSS NUMBERS OF SEQUENCES OVER FINITE ABELIAN GROUPS. 2024. ⟨hal-04546837⟩
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