1. ON LINEAR COMBINATIONS OF UNITS WITH BOUNDED COEFFICIENTS
- Author
-
Volker Ziegler and Jörg M. Thuswaldner
- Subjects
Discrete mathematics ,Degree (graph theory) ,Generalization ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Algebraic number field ,Mathematical proof ,01 natural sciences ,Integer ,Bounded function ,0101 mathematics ,Algebraic number ,Linear combination ,Mathematics - Abstract
Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units e i in a way that the coefficients a i ∈ℕ are bounded by n . The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.
- Published
- 2011