### Abstract

Let K be a number field of degree k and let O be an order in K. A generalized number system overO (GNS for short) is a pair (p, D) where p∈ O[x] is monic and D⊂ O is a complete residue system modulo p(0) containing 0. If each a∈ O[x] admits a representation of the form a≡∑j=0ℓ-1djxj(modp) with ℓ∈ N and d_{0}, … , d_{ℓ} _{-} _{1}∈ D then the GNS (p, D) is said to have the finiteness property. To a given fundamental domain F of the action of Z^{k} on R^{k} we associate a class GF:={(p,DF):p∈O[x]} of GNS whose digit sets D_{F} are defined in terms of F in a natural way. We are able to prove general results on the finiteness property of GNS in G_{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 F we characterize the finiteness property of (p(x± m) , D_{F}) 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.

Original language | English |
---|---|

Pages (from-to) | 681-704 |

Number of pages | 24 |

Journal | Monatshefte fur Mathematik |

Volume | 187 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2018 |

### Keywords

- Number field
- Number system
- Order
- Tiling

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'Number systems over orders'. Together they form a unique fingerprint.

## Cite this

*Monatshefte fur Mathematik*,

*187*(4), 681-704. https://doi.org/10.1007/s00605-018-1191-x