Related papers: A Cobham theorem for scalar multiplication

The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a…

Let $k,\ell\geq 2$ be two multiplicatively independent integers. Cobham's famous theorem states that a set $X\subseteq \mathbb{N}$ is both $k$-recognizable and $\ell$-recognizable if and only if it is definable in Presburger arithmetic.…

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually…

Let $(\alpha,\mathcal{N}_{\alpha})$ and $(\beta,\mathcal{N}_{\beta})$ be two canonical number systems for an imaginary quadratic number field $K$ such that $\alpha$ and $\beta$ are multiplicatively independent. We provide an effective lower…

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\mathbb{Q}(\beta) \rightarrow \mathbb{Q}(\alpha)$. The algorithm is particularly efficient if…

We prove that there exists a countable $\beta$-model in which, for all reals $X$ and $Y$, $X$ is definable from $Y$ if and only $X$ is hyperarithmetical in $Y$. We also obtain some related results and pose some related questions.

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 $\beta$…

Let $K$ be a subfield of $\mathbb{R}$. The theory of $\mathbb{R}$ viewed as an ordered $K$-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.

We prove that all algebraic bases $\beta$ allow an eventually periodic representations of the elements of $\mathbb Q(\beta)$ with a finite alphabet of digits $\mathcal A$. Moreover, the classification of bases allowing that those…

We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$, which we call $\boldsymbol{\beta}$-representations. In doing so, we generalize both representations…

In this note, we give an alternative proof of the following result. Let p, q >= 2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is…

Let $(q_{\alpha, n})_{n \geq 0}$ be the sequence of convergent denominators to the simple continued fraction expansion of $\alpha$. For certain specific choices of $\alpha$, this sequence is a Lehmer sequence. In this paper, we show that…

Feng and Wang showed that two homogeneous iterated function systems in $\mathbb{R}$ with multiplicatively independent contraction ratios necessarily have different attractors. In this paper, we extend this result to graph directed iterated…

We study rational numbers with purely periodic R\'enyi $\beta$-expansions. For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $\gamma(\beta)=1$, i.e., that all rational…

We study periodic expansions in positional number systems with a base $\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

In this paper we deal with a problem of Peth\H{o} related to existence of quartic algebraic integer $\alpha$ for which $$ \beta=\frac{4\alpha^4}{\alpha^4-1}-\frac{\alpha}{\alpha-1} $$ is a quadratic algebraic number. By studying rational…

We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha…

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is…

This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set…