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

Daniel Dombek; Zuzana Masáková; Tomáš Vávra

doi:10.5802/jtnb.922

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

Daniel Dombek ¹; Zuzana Masáková ²; Tomáš Vávra ²

¹ Faculty of Information Technology Czech Technical University in Prague Thákurova 9, 160 00 Praha 6 CZECH REPUBLIC
² Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Trojanova 13, 120 00 Praha 2 CZECH REPUBLIC

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 3, pp. 745-768.

Abstract
Résumé

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 $(\pm β)$ 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 ${0, 1, \dots, ⌊ β ⌋}$ is a $(- β)$ -integer, although the corresponding sequence of digits is forbidden as a $(- β)$ -expansion.

Nous étudions le développement des nombres réels en bases positive et negative, suivant les travaux de Rényi, et Ito & Sadahiro. Nous comparons les ensembles $ℤ_{β}^{+}$ et $ℤ_{- β}$ des nombres $β$ -entiers non-negatifs et $(- β)$ -entiers. Nous décrivons les bases $(\pm β)$ pour lesquelles $ℤ_{β}^{+}$ et $ℤ_{- β}$ sont codés par des mots infinis qui sont des points fixes de morphismes conjugés. De plus, nous démontrons que cela se produit précisement pour les nombres $β$ ayant la propriété suivante : toute combinaison linéaire de puissances non-négatives de la base $- β$ , à coefficients dans ${0, 1, \dots, ⌊ β ⌋}$ , correspond à un $(- β)$ -entier, même si la suite donnée de chiffres est interdite comme $(- β)$ -développement.

DOI: 10.5802/jtnb.922

Classification: 11A63, 68R15
Keywords: $(-\beta )$-expansion, $(-\beta )$-integer, confluent Parry number, spectrum, antimorphism, conjugacy.

Author's affiliations:

Daniel Dombek ¹; Zuzana Masáková ²; Tomáš Vávra ²

¹ Faculty of Information Technology Czech Technical University in Prague Thákurova 9, 160 00 Praha 6 CZECH REPUBLIC
² Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague Trojanova 13, 120 00 Praha 2 CZECH REPUBLIC

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     author = {Daniel Dombek and Zuzana Mas\'akov\'a and Tom\'a\v{s} V\'avra},
     title = {Confluent {Parry} numbers, their spectra,  and integers in positive- and negative-base  number systems},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {745--768},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {3},
     year = {2015},
     doi = {10.5802/jtnb.922},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.922/}
}

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Daniel Dombek; Zuzana Masáková; Tomáš Vávra. Confluent Parry numbers, their spectra,  and integers in positive- and negative-base  number systems. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 3, pp. 745-768. doi : 10.5802/jtnb.922. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.922/

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