Ing. Daniel Dombek, Ph.D.

Článek

10.5802/jtnb.922

Katedra aplikované matematiky

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.

Článek

10.1016/j.jnt.2014.09.029

Katedra aplikované matematiky

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.

Stať ve sborníku

Katedra aplikované matematiky

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.