Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems
Authors
Dombek, D.; Masáková, Z.; Vávra, T.
Year
2015
Published
J. Theor. Nombres Bordeaux. 2015, 27(3), 745-768. ISSN 1246-7405.
Type
Article
DOI
Departments
Annotation
In this paper we study the expansions of real numbers in positive and negative real base as introduced by Rényi, and Ito & Sadahiro, respectively. In particular, we compare the sets ℤ β + and ℤ -β of nonnegative β-integers and (-β)-integers. We describe all bases (±β) for which ℤ β + and ℤ -β can be coded by infinite words which are fixed points of conjugated morphisms, and consequently have the same language. Moreover, we prove that this happens precisely for β with another interesting property, namely that any linear combination of non-negative powers of the base -β with coefficients in {0,1,...,⌊β⌋} is a (-β)-integer, although the corresponding sequence of digits is forbidden as a (-β)-expansion.
On distinct unit generated fields that are totally complex
Authors
Dombek, D.; Masáková, Z.; Ziegler, V.
Year
2015
Published
Journal of Number Theory. 2015, 148 311-327. ISSN 0022-314X.
Type
Article
Departments
Annotation
We consider the problem of characterizing all number fields $K$
such that all algebraic integers $alphain K$ can be written as the sum of distinct
units of $K$. We extend a method due to Thuswaldner and Ziegler [12] that
previously did not work for totally complex fields and apply our results to the
case of totally complex quartic number fields.
Generating (+-beta)-integers by Conjugated Morphisms
Authors
Year
2013
Published
Local Proceedings of WORDS 2013. Turku: University of Turku, 2013. pp. 14-25. TUCS Lecture Notes. ISSN 1797-8831. ISBN 978-952-12-2939-8.
Type
Proceedings paper
Departments
Annotation
In this paper we study the expansions of real numbers in positive and negative real base. In particular, we consider the sets $mathbb{Z}_beta^+$ and $mathbb{Z}_{-beta}$ of nonnegative $beta$-integers and $(-beta)$-integers respectively. It is well known that, in numerous cases, $mathbb{Z}_{-beta}$ can be completely unrelated to $mathbb{Z}_beta^+$. We precisely describe all bases for which $mathbb{Z}_beta^+$ and $mathbb{Z}_{-beta}$ can be coded by infinite words with the same language.