Periodic representations in Salem bases

Vávra, Tomáš

doi:10.1007/s11856-021-2123-3

Cited by 11 publications

(7 citation statements)

References 18 publications

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“…Such idea was first presented in [3], where the authors show that any base β which is either a rational or an algebraic integer without conjugates on the unit circle admits a finite alphabet A of digits so that any element in the field Q(β) has a periodic (β, A)-representation. Vávra in [18] then extends this result to all algebraic bases, including Salem numbers.…”

Section: Commentsmentioning

confidence: 75%

Rewriting Rules for Arithmetics in Alternate Base Systems

Masáková

Pelantová

Katarína

2023

Lecture Notes in Computer Science

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We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the so-called alternate base B given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. on the set of numbers with eventually periodic B-expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic B-expansion.

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Section: Commentsmentioning

confidence: 75%

Rewriting Rules for Arithmetics in Alternate Base Systems

Masáková

Pelantová

Katarína

2023

Lecture Notes in Computer Science

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“…For p = 1, i.e., for classical numeration systems with one base β, the Erdős spectrum proved to be useful in different situations. Vávra in [22] used the spectrum to characterize complex bases β for which, with a suitably chosen digit set, every element of the field Q(β) has an eventually periodic representation. In [13] the question whether a numeration system with a complex base β and a digit set D ⊂ C allows a representation of any complex number is reformulated as the question whether the corresponding Erdős spectrum is relatively dense.…”

Section: Further Workmentioning

confidence: 99%

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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“…Schmidt conjectured that for real bases β that are Salem numbers, we still have that Per(β) = Q(β) ∩ [0, 1) [12]. Even though some work has been done, this problem is still open today, as well as the partial problem to know whether all Salem numbers are Parry numbers; see for instance [2,3,8,13]. Analogous questions can be asked in the alternate base framework.…”

Section: Open Questionsmentioning

confidence: 99%

On Periodic Alternate Base Expansions

Charlier¹,

Cisternino²,

Kreczman³

2022

Preprint

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For an alternate base β = (β0, . . . , βp−1), we show that if all rational numbers in the unit interval [0, 1) have periodic alternate expansions with respect to the p shifts of β, then the bases β0, . . . , βp−1 all belong to the extension field Q(β) where β is the product β0 • • • βp−1 and moreover, this product β must be either a Pisot number or a Salem number. We also prove the stronger statement that if the bases β0, . . . , βp−1 belong to Q(β) but the product β is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic β-expansion is nowhere dense in [0, 1). Moreover, in the case where the product β is a Pisot number and the bases β0, . . . , βp−1 all belong to Q(β), we prove that the set of points in [0, 1) having an ultimately periodic β-expansion is precisely the set Q(β) ∩ [0, 1). For the restricted case of Rényi real bases, i.e., for p = 1 in our setting, our method gives rise to an elementary proof of Schmidt's original result. We also give two applications of these results. First, if β = (β0, . . . , βp−1) is an alternate base such that the product β of the bases is a Pisot number and β0, . . . , βp−1 ∈ Q(β), then β is a Parry alternate base, meaning that the quasi-greedy expansions of 1 with respect to the p shifts of the base β are ultimately periodic. Second, we obtain a property of Pisot numbers, namely that if β is a Pisot number then β ∈ Q(β p ) for all integers p ≥ 1.

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Periodic representations in Salem bases

Cited by 11 publications

References 18 publications

Rewriting Rules for Arithmetics in Alternate Base Systems

Rewriting Rules for Arithmetics in Alternate Base Systems

Spectrum, algebraicity and normalization in alternate bases

On Periodic Alternate Base Expansions

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