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

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 nonnegativeβ-integers and (−β)-integers. We describe all bases (±β) for which ℤ β + and ℤ −β can...

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Bibliographic Details
Published in:Journal de theorie des nombres de bordeaux 2015-01, Vol.27 (3), p.745-768
Main Authors: DOMBEK, Daniel, MASÁKOVÁ, Zuzana, VÁVRA, Tomáš
Format: Article
Language:English
Subjects:
Alphabets
Coefficients
Integers
Morphisms
Numbers
Numeration
Power series coefficients
Real numbers
Words
Zero
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Summary: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 nonnegativeβ-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 forβwith another interesting property, namely that any linear combination of non-negative powers of the base −βwith coefficients in {0, 1, . . . , ⌊β⌋} is a (−β)-integer, although the corresponding sequence of digits is forbidden as a (−β)-expansion.
ISSN:1246-7405
2118-8572
DOI:10.5802/jtnb.922