On linear combinations of units with bounded coefficients and double-base digit expansions

Daniel Krenn; Jörg Thuswaldner; Volker Ziegler

doi:10.1007/s00605-012-0443-4

On linear combinations of units with bounded coefficients and double-base digit expansions

Mon Hefte Math. 2013;171(3-4):377-394. doi: 10.1007/s00605-012-0443-4. Epub 2012 Oct 6.

Authors

Daniel Krenn¹, Jörg Thuswaldner², Volker Ziegler³

Affiliations

¹ Institute of Optimisation and Discrete Mathematics (Math B), Graz University of Technology, Steyrergasse 30/II, 8010 Graz, Austria.
² Chair of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria.
³ Institute of Analysis and Computational Number Theory (Math A), Graz University of Technology, Steyrergasse 30/II, 8010 Graz, Austria.

Abstract

Let [Formula: see text] be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in [Formula: see text] is the sum of pairwise distinct units, if the unit equation [Formula: see text] has a non-trivial solution [Formula: see text]. We generalize this result and give applications to signed double-base digit expansions.

Keywords: Additive unit structure; Digit expansions; Unit sum number.