Related papers: Alphabets, rewriting trails and periodic represent…

For a complex number $x$, $\Vert x\Vert:=\min\{|x-m|:m\in\mathbb{Z}\}$. Let $k\geq 1$ be an integer, and $K$ be a number field. Let $\alpha_1,\ldots,\alpha_k$ be algebraic numbers with $|\alpha_i|\geq 1$ and let $d_i$ denotes the degree of…

We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of…

The beta-conjugates of a base of numeration $\beta > 1$, $\beta$ being a Parry number, were introduced by Boyd, in the context of the R\'enyi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are…

We show that every intermediate $\beta$-transformation is topologically conjugate to a greedy $\beta$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This…

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different…

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 real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the…

Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a pair of conjugate…

A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…

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…

We give criteria for finding the greedy $\beta$-expansion for $1$ for families of Salem numbers that approach a given Pisot number. We show that these expansions are related to the greedy expansion under the Pisot base. This expands on the…

We will introduce a family $\Gamma_\beta, 1 < \beta \in {\mathbb{R}}$ of infinite non-amenable discrete groups as an interpolation of the Higman-Thompson groups $V_n, 1 < n \in {\mathbb{N}}$ by using the topological full groups of the…

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations…

Let $ \mathbb{Q}\mathcal{E}_{\mathbb{Z}} $ be the set of power sums whose characteristic roots belong to $ \mathbb{Z} $ and whose coefficients belong to $ \mathbb{Q} $, i.e. $ G : \mathbb{N} \rightarrow \mathbb{Q} $ satisfies…

Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this…

We consider base-$\beta$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<\beta <a_0+1$ is the positive solution to $\beta^2 = a_0\beta + a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto…

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved…

Let $\Gamma$ be a rooted tree and let $t$ be a positive integer. We study algebraic invariants and properties of the path ideal generated by monomial corresponding to paths of length $(t-1)$ in $\Gamma$. In particular, we give a recursive…

We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental…

For $b>1$ and $\alpha\beta$ a string of two digits in base $b$, let $K_1$ be the subsum of the harmonic series with only those integers having exactly one occurrence of $\alpha\beta$. We obtain a theoretical representation of such $K_1$…