A characterization of rational elements by Lüroth-type series expansions in the p-adic number field and in the field of Laurent series over a finite field
A characterization of rational numbers by p-adic Sylvester series expansions
A logical alternative to the existing positional number system.
Addidtive representations of integers.
An exercise on Fibonacci representations
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 and related problems
Complexity of infinite words associated with beta-expansions
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 we show that . In the cases when or we show that the first difference of the complexity function takes value in for every , and consequently we determine...
Complexity of infinite words associated with beta-expansions
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.
Construction of normal numbers with respect to the Q-Cantor series expansion for certain Q
Corrigendum : “Complexity of infinite words associated with beta-expansions”
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
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.
Differences of multiple Fibonacci numbers.
Families of rational numbers with predictable Engel product expansions.
Farey fractions and sums over coprime pairs
Fraenkel's partition and Brown's decomposition.
Gaussian Integers
Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. 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
Generalized radix representations and dynamical systems II
Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
We investigate in a geometrical way the point sets of obtained by the -numeration that are the -integers where is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the -numeration, allowing to lift up the -integers to some points of the lattice ( degree of ) lying about the dominant eigenspace of the companion matrix of . When is in particular a Pisot number, this framework gives another proof of the fact that is...