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.
Categories: math.NT