Number systems over orders

LetKbe a number field of degree k and letObe an order inK. Ageneralized number system over O GNS for short) is a pair p, D) where p. O[x] is monic and D. O is a complete residue system modulo p0) containing 0. If each a. O[x] admits a representation of the form a = - 1 j= 0 dj x j mod p) with . N and d0,..., d - 1. D then the GNS p, D) is said to have the finiteness property. To a given fundamental domain F of the action of Zk on Rk we associate a class GF := {p, DF) : p. O[x]} of GNS whose digit sets DF are defined in terms of F in a natural way. We are able to prove general results on the finiteness property of GNS in GF by giving an abstract version of the well- known " dominant condition" on the absolute coefficient p0) of p. In particular, depending on mild conditions on the topology of F we characterize the finiteness property of px +/- m), DF) for fixed p and large m. N. Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.