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ON LINEAR COMBINATIONS OF UNITS WITH BOUNDED COEFFICIENTS

Part of: Elementary number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  31 May 2011

Jörg Thuswaldner  and
Volker Ziegler
Jörg Thuswaldner
Affiliation:
Chair of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria (email: Joerg.Thuswaldner@unileoben.ac.at)
Volker Ziegler
Affiliation:
Institute for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30/IV, A-8010 Graz, Austria (email: ziegler@math.tugraz.at)
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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 εi in a way that the coefficients ai∈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.

MSC classification

Secondary: 11R16: Cubic and quartic extensions 11R11: Quadratic extensions 11A63: Radix representation; digital problems
Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 247 - 262
Copyright
Copyright © University College London 2011

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