Number systems over orders

Let $\mathbb{K}$ be a number field of degree $k$ and let $\mathcal{O}$ be an order in $\mathbb{K}$. A \emph{generalized number system over $\mathcal{O}$} (GNS for short) is a pair $(p,\mathcal{D})$ where $p \in \mathcal{O}[x]$ is monic and $\mathcal{D}\subset\mathcal{O}$ is a complete residue system modulo $p(0)$ containing $0$. If each $a \in \mathcal{O}[x]$ admits a representation of the form $a \equiv \sum_{j =0}^{\ell-1} d_j x^j \pmod{p}$ with $\ell\in\mathbb{N}$ and $d_0,\ldots, d_{\ell-1}\in\mathcal{D}$ then the GNS $(p,\mathcal{D})$ is said to have the \emph{finiteness property}. To a given fundamental domain $\mathcal{F}$ of the action of $\mathbb{Z}^k$ on $\mathbb{R}^k$ we associate a class $\mathcal{G}_\mathcal{F} := \{ (p, D_\mathcal{F}) \;:\; p \in \mathcal{O}[x] \}$ of GNS whose digit sets $D_\mathcal{F}$ are defined in terms of $\mathcal{F}$ in a natural way. We are able to prove general results on the finiteness property of GNS in $\mathcal{G}_\mathcal{F}$ by giving an abstract version of the well-known "dominant condition" on the absolute coefficient $p(0)$ of $p$. In particular, depending on mild conditions on the topology of $\mathcal{F}$ we characterize the finiteness property of $(p(x\pm m), D_\mathcal{F})$ for fixed $p$ and large $m\in\mathbb{N}$. Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.