Related papers: On the structure of ($-\beta$)-integers

Given a totally finite ordered alphabet $ A $, endowing the set of words over $ A $ with the alternating lexicographic order, we define a new class of Lyndon words. We study the fundamental properties of the associated symbolic dynamical…

We consider lattices of regular sets of non negative integers, i.e. of sets definable in Presbuger arithmetic. We prove that if such a lattice is closed under decrement then it is also closed under many other functions: quotients by an…

Let $A$ be a set of positive integers. We define a positive integer $n$ as an $A$-practical number if every positive integer from the set $\left\{1,\ldots ,\sum_{d\in A, d\mid n}d\right\}$ can be written as a sum of distinct divisors of $n$…

Arbitrarily sparse sets A of integers are constructed with the property that every integer can be represented uniquely in the form n = a + a', where a and a' belong to the set A and a < a' or a = a'. Some related open problems are stated.

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

Let $\alpha,\beta \in \mathbb{R}_{>0}$ be such that $\alpha,\beta$ are quadratic and $\mathbb{Q}(\alpha)\neq \mathbb{Q}(\beta)$. Then every subset of $\mathbb{R}^n$ definable in both $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto \alpha x)$ and…

We study arithmetical aspects of Ito-Sadahiro number systems with negative base. We show that the bases $-\beta<-1$, where $\beta$ is zero of $x^3-mx^2-mx-m,\ m\in\mathbb N,$ possess the so-called finiteness property. For the Tribonacci…

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…

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to its connection to maximum matchings in a graph. In this paper, we first define $\theta$-barrier sets. Our definition of a $\theta$-barrier set…

An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure…

We find the precise rate at which the empirical measure associated to a $\beta$-ensemble converges to its limiting measure. In our setting the $\beta$-ensemble is a random point process on a compact complex manifolds distributed according…

The (partially) ordered set of the non-trivial zeros of the zeta function with positive imaginary parts is considered. The order is the coordinatewise order inherited from $\mathbb{C}$. Some interesting properties regarding the minimal…

A set A of positive integers is called a h-bais of [0,n] if each integer in [0,n]is a sum of no more than h members of A. In this paper, we will give a new construction for h-basis.

Let $\beta$ be any permutation on $n$ symbols and let $c(k, \beta)$ be the number of permutations that $k$-commute with $\beta$. The cycle type of a permutation $\beta$ is a vector $(c_1, \dots, c_n)$ such that $\beta$ has exactly $c_i$…

Let $\Phi$ be a subset of the simple roots of a (possibly non-reduced) abstract root system $\Sigma$, and let $\lambda \in \Sigma$. We define the $\Phi$-string of $\lambda$ as the set of elements in $\Sigma \cup \{0\}$ of the form $\lambda…

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…

We classify $R$-spaces that admit a certain natural $\Gamma$-symmetric structure. We further determine the maximal antipodal sets of these structures.

A relational structure is called reversible iff every bijective endomorphism of that structure is an automorphism. We give several equivalents of that property in the class of disconnected binary structures and some its subclasses. For…

Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under…

The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite. Zeta-dimension serves as a fractal dimension on the positive integers that extends…