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Acta Arithmetica

Acta Arithmetica

### A logical alternative to the existing positional number system.

Southwest Journal of Pure and Applied Mathematics [electronic only]

Manuscripta mathematica

### An exercise on Fibonacci representations

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.

### Characterization of the unique expansions $1={\sum }_{i=1}^{\infty }{q}^{-{n}_{i}}$ and related problems

Bulletin de la Société Mathématique de France

### 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...

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

RAIRO - Theoretical Informatics and Applications

We study the complexity of the infinite word uβ associated with the Rényi expansion of 1 in an irrational base β > 1. When β is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity C(n) = n + 1. For β such that dβ(1) = t1t2...tm is finite we provide a simple description of the structure of special factors of the word uβ. When tm=1 we show that C(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1or t1 > max{t2,...,tm-1} we show that the first difference of...

### Connectedness of number theoretic tilings.

Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]

Acta Arithmetica

### 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 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.

Integers

### Families of rational numbers with predictable Engel product expansions.

Journal of Integer Sequences [electronic only]

Acta Arithmetica

Integers

### Gaussian Integers

Formalized Mathematics

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers . We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

### Généralisation d'un résultat de Loxton et van der Poorten

Journal de théorie des nombres de Bordeaux

Acta Arithmetica

### Geometric study of the beta-integers for a Perron number and mathematical quasicrystals

Journal de Théorie des Nombres de Bordeaux

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${ℤ}_{\beta }\subset ℤ\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${ℤ}^{m}$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact that  ${ℤ}_{\beta }$  is...

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