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arXiv · 1402.4314

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

Daniel Dombek·Zuzana Masáková·Tomáš Vávra
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Abstract

In this paper we study the expansions of real numbers in positive and negative real base as introduced by R\'enyi, and Ito & Sadahiro, respectively. In particular, we compare the sets $\mathbb{Z}_\beta^+$ and $\mathbb{Z}_{-\beta}$ of nonnegative $\beta$-integers and $(-\beta)$-integers. We describe all bases $(\pm\beta)$ for which $\mathbb{Z}_\beta^+$ and $\mathbb{Z}_{-\beta}$ 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 $\beta$ with another interesting property, namely that any integer linear combination of non-negative powers of the base $-\beta$ with coefficients in $\{0,1,\dots,\lfloor\beta\rfloor\}$ is a $(-\beta)$-integer, although the corresponding sequence of digits is forbidden as a $(-\beta)$-integer.

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Daniel Dombek, Zuzana Masáková, Tomáš Vávra. 2014-02-18. Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems. https://arxiv.org/abs/1402.4314

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