Expansions in Cantor real bases

Charlier, Émilie; Cisternino, Célia

doi:10.1007/s00605-021-01598-6

Cited by 13 publications

(62 citation statements)

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“…Theorem 4 (Charlier and Cisternino [6]). An infinite word a over N is the β-expansion of some real number in the interval [0, 1) if and only if…”

Section: 3mentioning

confidence: 99%

“…Representations of real numbers involving more than one base simultaneously and independently aroused the interest of other mathematicians [4,14,16,23]. But so far, most of the research was concentrated on the combinatorial properties of these representations as in [6] and not on their dynamical or algebraic properties which are studied respectively in [7] and in this paper.…”

Section: Introductionmentioning

confidence: 96%

“…. , β p−1 ) was proved in [6]. Namely, it is proved that the alternate base system defines a sofic βshift if and only if each of the p greedy β (i) -expansions of 1 is eventually periodic where β (i) = (β i , .…”

Section: Introductionmentioning

confidence: 99%

“…Alternate bases are particular cases of Cantor real bases, which were introduced by the first two authors in [6] and then studied in [7,8]. A Cantor real base is a sequence β = (β n ) n∈N of real numbers greater than 1 such that +∞ n=0 β n = +∞.…”

Section: Introductionmentioning

confidence: 99%

“…Two particular β-representations of x ∈ [0, 1] can be obtained thanks to the greedy and lazy algorithms. In [6], the classical combinatorial results of the greedy β-expansions were generalized to the framework of Cantor re al bases. In [8], the lazy combinatorial properties of Cantor real bases were investigated.…”

Section: Introductionmentioning

confidence: 99%

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Spectrum, algebraicity and normalization in alternate bases

Charlier¹,

Cisternino²,

Masáková³

et al. 2022

Preprint

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The first aim of this article is to give information about the algebraic properties of alternate bases β = (β0, . . . , βp−1) determining sofic systems. We show that a necessary condition is that the product δ = p−1 i=0 βi is an algebraic integer and all of the bases β0, . . . , βp−1 belong to the algebraic field Q(δ). On the other hand, we also give a sufficient condition: if δ is a Pisot number and β0, . . . , βp−1 ∈ Q(δ), then the system associated with the alternate base β = (β0, . . . , βp−1) is sofic. The second aim of this paper is to provide an analogy of Frougny's result concerning normalization of real bases representations. We show that given an alternate base β = (β0, . . . , βp−1) such that δ is a Pisot number and β0, . . . , βp−1 ∈ Q(δ), the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. The spectrum of a real number δ > 1 and an alphabet A ⊂ Z was introduced by Erdős et al. For our purposes, we use a generalized concept with δ ∈ C and A ⊂ C and study its topological properties.

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“…Theorem 4 (Charlier and Cisternino [6]). An infinite word a over N is the β-expansion of some real number in the interval [0, 1) if and only if…”

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Spectrum, algebraicity and normalization in alternate bases

Charlier¹,

Cisternino²,

Masáková³

et al. 2022

Preprint

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A Full Characterization of Bertrand Numeration Systems

Charlier

Cisternino

Stipulanti

2022

Developments in Language Theory

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We consider Cantor real numeration system as a frame in which every nonnegative real number has a positional representation. The system is defined using a bi-infinite sequence B = (βn) n∈Z of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbolic sequence in general over the alphabet N. We show that this sequence is S-adic. We focus on alternate base systems, where the sequence B of bases is periodic and characterize alternate bases B, in which B-integers can be coded using a symbolic sequence vB over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that vB is a fixed point of their composition. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base B is a constant sequence.

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Alternate Base Numeration Systems

Charlier

2023

Lecture Notes in Computer Science

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Alternate base numeration systems generalize real base numeration systems as defined by Renyi. Real numbers are represented using a finite number of bases periodically. Such systems naturally appear when considering linear numeration systems without a dominant root. As it happens, many classical results generalize to these numeration systems with multiple bases but some don't. This is a survey of the work done so far concerning combinatorial, algebraic and dynamical aspects. This study has been led in collaboration with several co-authors :

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Expansions in Cantor real bases

Cited by 13 publications

References 10 publications

Spectrum, algebraicity and normalization in alternate bases

Spectrum, algebraicity and normalization in alternate bases

A Full Characterization of Bertrand Numeration Systems

Alternate Base Numeration Systems

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