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### Palindromic complexity of infinite words associated with non-simple Parry numbers

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the palindromic complexity of infinite words ${u}_{\beta }$, the fixed points of the substitution over a binary alphabet, $\varphi \left(0\right)={0}^{a}1$, $\varphi \left(1\right)={0}^{b}1$, with $a-1\ge b\ge 1$, which are canonically associated with quadratic non-simple Parry numbers $\beta$.

### Palindromic complexity of infinite words associated with non-simple Parry numbers

RAIRO - Theoretical Informatics and Applications

We study the palindromic complexity of infinite words , the fixed points of the substitution over a binary alphabet, , , with , which are canonically associated with quadratic non-simple Parry numbers .

### Arithmetics in numeration systems with negative quadratic base

Kybernetika

We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta >1$ of polynomials ${x}^{2}-mx-n$, $m\ge n\ge 1$, and show that in this case the set $\mathrm{Fin}\left(-\beta \right)$ of finite $\left(-\beta \right)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta =\tau =\frac{1}{2}\left(1+\sqrt{5}\right)$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits...

### Morphisms fixing words associated with exchange of three intervals

RAIRO - Theoretical Informatics and Applications

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.

### Integers with a maximal number of Fibonacci representations

RAIRO - Theoretical Informatics and Applications

We study the properties of the function which determines the number of representations of an integer as a sum of distinct Fibonacci numbers . We determine the maximum and mean values of for .

### Complexity of infinite words associated with beta-expansions

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the complexity of the infinite word ${u}_{\beta }$ associated with the Rényi expansion of $1$ in an irrational base $\beta >1$. When $\beta$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ\left(n\right)=n+1$. For $\beta$ such that ${d}_{\beta }\left(1\right)={t}_{1}{t}_{2}\cdots {t}_{m}$ is finite we provide a simple description of the structure of special factors of the word ${u}_{\beta }$. When ${t}_{m}=1$ we show that $ℂ\left(n\right)=\left(m-1\right)n+1$. In the cases when ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}$ or ${t}_{1}>max\left\{{t}_{2},\cdots ,{t}_{m-1}\right\}$ we show that the first difference of the complexity function $ℂ\left(n+1\right)-ℂ\left(n\right)$ takes value in $\left\{m-1,m\right\}$ for every $n$, and consequently we determine...

### Integers with a maximal number of Fibonacci representations

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the properties of the function $R\left(n\right)$ which determines the number of representations of an integer $n$ as a sum of distinct Fibonacci numbers ${F}_{k}$. We determine the maximum and mean values of $R\left(n\right)$ for ${F}_{k}\le n<{F}_{k+1}$.

### Corrigendum : “Complexity of infinite words associated with beta-expansions”

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny et al. (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny et al. (2004) use it.

### Corrigendum: Complexity of infinite words associated with beta-expansions

RAIRO - Theoretical Informatics and Applications

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny (2004) use it.

### Complexity of infinite words associated with beta-expansions

RAIRO - Theoretical Informatics and Applications

We study the complexity of the infinite word associated with the Rényi expansion of in an irrational base . When is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity . For such that is finite we provide a simple description of the structure of special factors of the word . When =1 we show that . In the cases when or max} we show that the first difference of the complexity function takes value in for every , and consequently we determine the complexity...

### Greedy and lazy representations in negative base systems

Kybernetika

We consider positional numeration systems with negative real base $-\beta$, where $\beta >1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $\left(-\beta \right)$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base ${\beta }^{2}$ with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...

### Combinatorial properties of infinite words associated with cut-and-project sequences

Journal de théorie des nombres de Bordeaux

The aim of this article is to study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We consider here the cut-and-project scheme in two dimensions with general orientation of the projecting subspaces. We prove that a cut-and-project sequence arising in such a setting has always either two or three types of distances between adjacent points. A cut-and-project sequence thus determines in a natural way a symbolic sequence (infinite word)...

### Palindromic complexity of infinite words associated with simple Parry numbers

Annales de l’institut Fourier

A simple Parry number is a real number $\beta >1$ such that the Rényi expansion of $1$ is finite, of the form ${d}_{\beta }\left(1\right)={t}_{1}\cdots {t}_{m}$. We study the palindromic structure of infinite aperiodic words ${u}_{\beta }$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word ${u}_{\beta }$ contains infinitely many palindromes if and only if ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}\ge {t}_{m}$. Numbers $\beta$ satisfying this condition are the so-called Pisot numbers. If ${t}_{m}=1$ then ${u}_{\beta }$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then $𝒫\left(n+1\right)+𝒫\left(n\right)=𝒞\left(n+1\right)-𝒞\left(n\right)+2$, where...

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