Ing. Daniel Dombek, Ph.D.

Publikace

Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

Autoři
Dombek, D.; Masáková, Z.; Vávra, T.
Rok
2015
Publikováno
J. Theor. Nombres Bordeaux. 2015, 27(3), 745-768. ISSN 1246-7405.
Typ
Článek
Anotace
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

Autoři
Dombek, D.; Masáková, Z.; Ziegler, V.
Rok
2015
Publikováno
Journal of Number Theory. 2015, 148 311-327. ISSN 0022-314X.
Typ
Článek
Anotace
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

Autoři
Rok
2013
Publikováno
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.
Typ
Stať ve sborníku
Anotace
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.