### 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

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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.

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 $\u2102\left(n\right)=n+1$. 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 $\u2102\left(n\right)=(m-1)n+1$. In the cases when ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}$ or ${t}_{1}\>max\{{t}_{2},\cdots ,{t}_{m-1}\}$ we show that the first difference of the complexity function $\u2102(n+1)-\u2102\left(n\right)$ takes value in $\{m-1,m\}$ for every $n$, and consequently we determine...

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...

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.

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.

We investigate in a geometrical way the point sets of $\mathbb{R}$ obtained by the $\beta $-numeration that are the $\beta $-integers ${\mathbb{Z}}_{\beta}\subset \mathbb{Z}\left[\beta \right]$ 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 ${\mathbb{Z}}^{m}$ ($m=$ 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 ${\mathbb{Z}}_{\beta}$ is...