Number systems over orders

Let $$ be a number field of degree k and let $$ be an order in $$ . A generalized number system over $$ (GNS for short) is a pair $$ where $$ is monic and $$ is a complete residue system modulo p(0) containing 0. If each $$ admits a representation of the form $$ with $$ and $$ then the GNS $$ is said to have the finiteness property. To a given fundamental domain $$ of the action of $$ on $$ we associate a class $$ of GNS whose digit sets $$ are defined in terms of $$ in a natural way. We are able to prove general results on the finiteness property of GNS in $$ by giving an abstract version of the well-known "dominant condition" on the absolute coefficient p(0) of p. In particular, depending on mild conditions on the topology of $$ we characterize the finiteness property of $$ for fixed p and large $$ . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.