Sets of Cross Numbers of Sequences over Finite Abelian Groups

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.