Related papers: Combinatorial and Arithmetical Properties of Infin…

A simple Parry number is a real number \beta>1 such that the R\'enyi expansion of 1 is finite, of the form d_\beta(1)=t_1...t_m. We study the palindromic structure of infinite aperiodic words u_\beta that are the fixed point of a…

We study the palindromic complexity of infinite words $u_\beta$, the fixed points of the substitution over a binary alphabet, $\phi(0)=0^a1$, $\phi(1)=0^b1$, with $a-1\geq b\geq 1$, which are canonically associated with quadratic non-simple…

We consider numeration systems with base $\beta$ and $-\beta$, for quadratic Pisot numbers $\beta$ and focus on comparing the combinatorial structure of the sets $\Z_\beta$ and $\Z_{-\beta}$ of numbers with integer expansion in base…

In this paper, we study the critical exponent of infinite words $\ubeta$ coding $\beta$-integers for $\beta$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the…

The factor complexity of the infinite word $\ubeta$ canonically associated to a non-simple Parry number $\beta$ is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a…

Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of…

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic''…

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$…

This contribution is devoted to the study of positional numeration systems with negative base introduced by Ito and Sadahiro in 2009, called (-\beta)-expansions. We give an admissibility criterion for more general case of…

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…

We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmod{\alpha}$ has the solution $\beta$ with $0 < \beta < \alpha$ if and only if $\alpha/\beta$ has a continued fraction expansion with sequence of…

We prove that for a finite set of points $X$ in the projective $n$-space over any field, the Betti number $\beta_{n,n+1}$ of the coordinate ring of $X$ is non-zero if and only if $X$ lies on the union of two planes whose sum of dimension is…

Let $\pi(A)$, $\xi(A)$ and $\nu(A)$, respectively, denote the number of positive, zero and negative eigenvalues of the matrix $A$. Then the triplet $(\pi(A), \xi(A), \nu(A))$ is called the \emph{inertia} of $A$ and is denoted by…

In this paper we study the expansions of real numbers in positive and negative real base as introduced by R\'enyi, and Ito & Sadahiro, respectively. In particular, we compare the sets $\mathbb{Z}_\beta^+$ and $\mathbb{Z}_{-\beta}$ of…

Consider the minimal $\beta$-shift containing the shift space generated by given Sturmian word. In this paper we characterize such $\beta$ and investigate its combinatorial, dynamical and topological properties and prove that such $\beta$…

The lazy algorithm for a real base $\beta$ is generalized to the setting of Cantor bases $\boldsymbol{\beta}=(\beta_n)_{n\in \mathbb{N}}$ introduced recently by Charlier and the author. To do so, let $x_{\boldsymbol{\beta}}$ be the greatest…

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $\beta$ such that $|\beta|>1$, the number $[\![\boldsymbol{u} ]\!]_\beta:=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$…

For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the…

We consider finite $\beta$-ensembles $\mathcal X_{n,\beta}^{\mathbb F}$ with $n$ points on $\mathbb F$, where $\mathbb F$ denotes either the real line or the complex plane. Let $U$ be a bounded subset of $ \mathbb F$ such that $\partial U$…

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…