Number systems over orders

Let [Formula: see text] be a number field of degree k and let [Formula: see text] be an order in [Formula: see text] . A generalized number system over [Formula: see text] (GNS for short) is a pair [Formula: see text] where [Formula: see text] is monic and [Formula: see text] is a complete residue system modulo p(0) containing 0. If each [Formula: see text] admits a representation of the form [Formula: see text] with [Formula: see text] and [Formula: see text] then the GNS [Formula: see text] is said to have the finiteness property. To a given fundamental domain [Formula: see text] of the action of [Formula: see text] on [Formula: see text] we associate a class [Formula: see text] of GNS whose digit sets [Formula: see text] are defined in terms of [Formula: see text] in a natural way. We are able to prove general results on the finiteness property of GNS in [Formula: see text] 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 [Formula: see text] we characterize the finiteness property of [Formula: see text] for fixed p and large [Formula: see text] . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.