arXiv subjects

Célia Cisternino

Publications and source records attributed to Célia Cisternino.

12 recordsLinked to original sources

arXiv2023

Substitutions and Cantor real numeration systems

We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence $\Beta=(β_n)_{n\in\Z}$ of real numbers greater than one. We introduce the set of $\Beta$-integers and code the sequence of gaps between consecutive $\Beta$-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 $\Beta$ of bases is periodic and characterize alternate bases $\Beta$, in which $\Beta$-integers can be coded using a symbolic sequence $v_{\Beta}$ over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that $v_\Beta$ 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 $\Beta$ is a constant sequence.

math.CO↗

arXiv2023

On Periodic Alternate Base Expansions

For an alternate base $\boldsymbolβ=(β_0,\ldots,β_{p-1})$, we show that if all rational numbers in the unit interval $[0,1)$ have periodic expansions with respect to the $p$ shifts of $\boldsymbolβ$, then the bases $β_0,\ldots,β_{p-1}$ all belong to the extension field $\mathbb Q(β)$ where $β$ is the product $β_0\cdotsβ_{p-1}$ and moreover, this product $β$ must be either a Pisot or Salem number. We also prove the stronger statement that if the bases $β_0,\ldots,β_{p-1}$ belong to $\mathbb Q(β)$ but the product $β$ is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic $\boldsymbolβ$-expansion is nowhere dense in $[0,1)$. Moreover, in the case where the product $β$ is a Pisot number and the bases $β_0,\ldots,β_{p-1}$ all belong to $\mathbb Q(β)$, we prove that the set of points in $[0,1)$ having an ultimately periodic $\boldsymbolβ$-expansion is precisely the set $\mathbb Q(β)\cap[0,1)$. 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. Therefore, even though our results generalize those of Schmidt, our proofs should not be seen as generalizations of Schmidt's original arguments but as an original method in the generalized framework of alternate bases, which moreover gives a new elementary proof of Schmidt's results from 1980. As an application of our results, we show that if $\boldsymbolβ=(β_0,\ldots,β_{p-1})$ is an alternate base such that the product $β$ of the bases is a Pisot number and $β_0,\ldots,β_{p-1}\in\mathbb Q(β)$, then $\boldsymbolβ$ is a Parry alternate base, meaning that the quasi-greedy expansions of $1$ with respect to the $p$ shifts of the base $\boldsymbolβ$ are ultimately periodic.

math.NT↗

arXiv2022

A full characterization of Bertrand numeration systems

Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base $β$. However, the given condition turns to be not necessary. Hence, the goal of this paper is to provide a correction of Bertrand-Mathis' result. The main difference arises when $β$ is a Parry number, in which case are derived two associated Bertrand numeration systems. Along the way, we define a non-canonical $β$-shift and study its properties analogously to those of the usual canonical one.

math.CO↗

arXiv2022

Spectrum, algebraicity and normalization in alternate bases

The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbolβ=(β_0,\dots,β_{p-1})$ determining sofic systems. We show that a necessary condition is that the product $δ=\prod_{i=0}^{p-1}β_i$ is an algebraic integer and all of the bases $β_0,\ldots,β_{p-1}$ belong to the algebraic field ${\mathbb Q}(δ)$. On the other hand, we also give a sufficient condition: if $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, then the system associated with the alternate base $\boldsymbolβ=(β_0,\dots,β_{p-1})$ 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 $\boldsymbolβ=(β_0,\dots,β_{p-1})$ such that $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, 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 $A\subset {\mathbb Z}$ was introduced by Erdős et al. For our purposes, we use a generalized concept with $δ\in{\mathbb C}$ and $A\subset{\mathbb C}$ and study its topological properties.

math.CO↗

arXiv2022

Combinatorial properties of lazy expansions in Cantor real bases

The lazy algorithm for a real base $β$ is generalized to the setting of Cantor bases $\boldsymbolβ=(β_n)_{n\in \mathbb{N}}$ introduced recently by Charlier and the author. To do so, let $x_{\boldsymbolβ}$ be the greatest real number that has a $\boldsymbolβ$-representation $a_0a_1a_2\cdots$ such that each letter $a_n$ belongs to $\{0,\ldots,\lceil β_n \rceil -1\}$. This paper is concerned with the combinatorial properties of the lazy $\boldsymbolβ$-expansions, which are defined when $x_{\boldsymbolβ}<+\infty$. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of $x_{\boldsymbolβ}$ is proved. First, it is shown that the lazy $\boldsymbolβ$-expansions are obtained by "flipping" the digits of the greedy $\boldsymbolβ$-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy $\boldsymbolβ$-expansions of some real number in $(x_{\boldsymbolβ}-1,x_{\boldsymbolβ}]$ is proved. Moreover, the lazy $\boldsymbolβ$-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis' theorem in the lazy framework is proved: the lazy $\boldsymbolβ$-shift is sofic if and only if all quasi-lazy $\boldsymbolβ^{(i)}$-expansions of $x_{\boldsymbolβ^{(i)}}-1$ are ultimately periodic, where $\boldsymbolβ^{(i)}$ is the $i$-th shift of the alternate base $\boldsymbolβ$.

math.CO↗

arXiv2021

Dynamical behavior of alternate base expansions

We generalize the greedy and lazy $β$-transformations for a real base $β$ to the setting of alternate bases $\boldsymbolβ=(β_0,\ldots,β_{p-1})$, which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_\boldsymbolβ$ and $L_\boldsymbolβ$ respectively, can be iterated in order to generate the digits of the greedy and lazy $\boldsymbolβ$-expansions of real numbers. The aim of this paper is to describe the dynamical behaviors of $T_\boldsymbolβ$ and $L_\boldsymbolβ$. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the $p$-Lebesgue measure) $T_\boldsymbolβ$-invariant measure. We then show that this unique measure is in fact equivalent to the $p$-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $\frac{1}{p}\log(β_{p-1}\cdots β_0)$. We then express the density of this measure and compute the frequencies of letters in the greedy $\boldsymbolβ$-expansions. We obtain the dynamical properties of $L_\boldsymbolβ$ by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $β$-shift. Finally, we show that the $\boldsymbolβ$-expansions can be seen as $(β_{p-1}\cdots β_0)$-representations over general digit sets and we compare both frameworks.

math.DS↗

arXiv2021

Expansions in Cantor real bases

We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\boldsymbolβ=(β_n)_{n\in\mathbb{N}}$, which we call $\boldsymbolβ$-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of $\boldsymbolβ$-representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry's theorem characterizing sequences of nonnegative integers that are the greedy $\boldsymbolβ$-representations of some real number in the interval $[0,1)$. We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the $\boldsymbolβ$-shift is sofic if and only if all quasi-greedy $\boldsymbolβ^{(i)}$-expansions of $1$ are ultimately periodic, where $\boldsymbolβ^{(i)}$ is the $i$-th shift of the Cantor real base $\boldsymbolβ$.

math.CO↗

arXiv2021

Robustness of Pisot-regular sequences

We consider numeration systems based on a $d$-tuple $\mathbf{U}=(U_1,\ldots,U_d)$ of sequences of integers and we define $(\mathbf{U},\mathbb{K})$-regular sequences through $\mathbb{K}$-recognizable formal series, where $\mathbb{K}$ is any semiring. We show that, for any $d$-tuple $\mathbf{U}$ of Pisot numeration systems and any commutative semiring $\mathbb{K}$, this definition does not depend on the greediness of the $\mathbf{U}$-representations of integers. The proof is constructive and is based on the fact that the normalization is realizable by a $2d$-tape finite automaton. In particular, we use an ad hoc operation mixing a $2d$-tape automaton and a $\mathbb{K}$-automaton in order to obtain a new $\mathbb{K}$-automaton.

math.CO↗

arXiv2020

Regular sequences and synchronized sequences in abstract numeration systems

The notion of $b$-regular sequences was generalized to abstract numeration systems by Maes and Rigo in 2002. Their definition is based on a notion of $\mathcal{S}$-kernel that extends that of $b$-kernel. However, this definition does not allow us to generalize all of the many characterizations of $b$-regular sequences. In this paper, we present an alternative definition of $\mathcal{S}$-kernel, and hence an alternative definition of $\mathcal{S}$-regular sequences, which enables us to use recognizable formal series in order to generalize most (if not all) known characterizations of $b$-regular sequences to abstract numeration systems. We then give two characterizations of $\mathcal{S}$-automatic sequences as particular $\mathcal{S}$-regular sequences. Next, we present a general method to obtain various families of $\mathcal{S}$-regular sequences by enumerating $\mathcal{S}$-recognizable properties of $\mathcal{S}$-automatic sequences. As an example of the many possible applications of this method, we show that, provided that addition is $\mathcal{S}$-recognizable, the factor complexity of an $\mathcal{S}$-automatic sequence defines an $\mathcal{S}$-regular sequence. In the last part of the paper, we study $\mathcal{S}$-synchronized sequences. Along the way, we prove that the formal series obtained as the composition of a synchronized relation and a recognizable series is recognizable. As a consequence, the composition of an $\mathcal{S}$-synchronized sequence and a $\mathcal{S}$-regular sequence is shown to be $\mathcal{S}$-regular. All our results are presented in an arbitrary dimension $d$ and for an arbitrary semiring $\mathbb{K}$.

math.CO↗

arXiv2019

Minimal automaton for multiplying and translating the Thue-Morse set

The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word ${\tt abbabaabbaababba\cdots}$, which is the fixed point starting with ${\tt a}$ of the word morphism ${\tt a\mapsto ab,b\mapsto ba}$. The numbers in $\mathcal{T}$ are commonly called the {\em evil numbers}. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e.\ the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m-1\}$. The proposed method is general for any $b$-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given $2^p$-recognizable set is equal to a set of the form $m\mathcal{T}+r$.

cs.FL↗

arXiv2019

State Complexity of the Multiples of the Thue-Morse Set

The Thue-Morse set T is the set of those non-negative integers whose binary expansions have an even number of 1. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word abbabaabbaababba..., which is the fixed point starting with a of the word morphism sending a to ab and b to ba. The numbers in T are sometimes called the evil numbers. We obtain an exact formula for the state complexity (i.e. the number of states of its minimal automaton) of the multiplication by a constant of the Thue-Morse set with respect to any integer base b which is a power of 2. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all 2^p-expansions of the set mT for any positive integers m and p. The used method is general for any b-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given 2^p-recognizable set is equal to some multiple of the Thue-Morse set.

cs.FL↗

arXiv2019

State complexity of the multiples of the Thue-Morse set

The Thue-Morse set is the set of those nonnegative integers whose binary expansions have an even number of $1$. We obtain an exact formula for the state complexity of the multiplication by a constant of the Thue-Morse set $\mathcal{T}$ with respect with any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}$ for any positive integers $m$ and $p$.

cs.FL↗