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Émilie Charlier

Publications and source records attributed to Émilie Charlier.

17 recordsLinked to original sources
arXiv2026

Parry condition, existence and uniqueness of alternate bases

Alternate bases are a numeration system that generalizes the Rényi numeration system. It is common in this context to construct examples or counter-examples by specifying the expansions of $1$ in the desired system. While it is easy to show when a system with given expansions of $1$ exists in the Rényi case, the same is not true in the alternate case. In this article, we establish conditions for given words to be the expansions of $1$ in the alternate case. To do so, we use a fixed point theorem on matrices defined from the expansions and obtain the elements of the base from the components of the fixed point. We also obtain a partial result for the uniqueness of such a base. In the latter parts of the article, we use similar techniques to prove the existence of bases with a given sequence of $B$-integers.

math.NT↗
arXiv2025

Numeration systems without a dominant root and regularity

Positional numeration systems are a large family of numeration systems used to represent natural numbers. Whether the set of all representations forms a regular language or not is one of the most important questions that can be asked of such a system. This question was investigated in a 1998 article by Hollander. Central to his analysis is a property linking positional numeration systems and Rényi numeration systems, which use a real base to represent real numbers. However, this link only exists when the initial numeration system has a dominant root, which is not a necessary condition for regularity. In this article, we show a more general link between positional numeration systems and alternate base numeration systems, a family generalizing Rényi systems. We then take advantage of this link to provide a full characterization of those numeration systems that generate a regular language. We also discuss the effectiveness of our method, and comment Hollander's results and conjecture in the light of ours.

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arXiv2025

Computing Expansions in Infinitely Many Cantor Real Bases via a Single Transducer

Representing real numbers using convenient numeration systems (integer bases, $β$-numeration, Cantor bases, etc.) has been a longstanding mathematical challenge. This paper focuses on Cantor real bases and, specifically, on automatic Cantor real bases and the properties of expansions of real numbers in this setting. We develop a new approach where a single transducer associated with a fixed real number $r$, computes the $\mathbf{B}$-expansion of $r$ but for an infinite family of Cantor real bases $\mathbf{B}$ given as input. This point of view contrasts with traditional computational models for which the numeration system is fixed. Under some assumptions on the finitely many Pisot numbers occurring in the Cantor real base, we show that only a finite part of the transducer is visited. We obtain fundamental results on the structure of this transducer and on decidability problems about these expansions, proving that for certain classes of Cantor real bases, key combinatorial properties such as greediness of the expansion or periodicity can be decided algorithmically.

math.NT↗
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↗
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↗
arXiv2020

Recurrence along directions in multidimensional words

In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A $d$-dimensional word is called \emph{uniformly recurrent} if for all $(s_1,\ldots,s_d)\in\mathbb{N}^d$ there exists $n\in\mathbb{N}$ such that each block of size $(n,\ldots,n)$ contains the prefix of size $(s_1,\ldots,s_d)$. We are interested in a modification of this property. Namely, we ask that for each rational direction $(q_1,\ldots,q_d)$, each rectangular prefix occurs along this direction in positions $\ell(q_1,\ldots,q_d)$ with bounded gaps. Such words are called \emph{uniformly recurrent along all directions}. We provide several constructions of multidimensional words satisfying this condition, and more generally, a series of four increasingly stronger conditions. In particular, we study the uniform recurrence along directions of multidimentional rotation words and of fixed points of square morphisms.

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↗
arXiv2019

Nyldon words

The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.

math.CO↗
arXiv2014

Infinite Self-Shuffling Words

In this paper we introduce and study a new property of infinite words: An infinite word $x\in A^\mathbb{N}$, with values in a finite set $A$, is said to be $k$-self-shuffling $(k\geq 2)$ if $x$ admits factorizations: $x=\prod_{i=0}^\infty U_i^{(1)}\cdots U_i^{(k)}=\prod_{i=0}^\infty U_i^{(1)}=\cdots =\prod_{i=0}^\infty U_i^{(k)}$. In other words, there exists a shuffle of $k$-copies of $x$ which produces $x$. We are particularly interested in the case $k=2$, in which case we say $x$ is self-shuffling. This property of infinite words is shown to be an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue-Morse word and all Sturmian words of intercept $0<ρ<1$ (while those of intercept $ρ=0$ are not self-shuffling). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the {\it stepping stone model}. One important feature of self-shuffling words stems from its morphic invariance, which provides a useful tool for showing that one word is not the morphic image of another. The notion of self-shuffling has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.

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