Related papers: Arithmetics in numeration systems with negative qu…
In the present paper we want to focus on this dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These…
In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…
We study the values of the zeta-function of the root system of type $G_2$ at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include…
We establish the asymptotic expansion in $\beta$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling…
Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number (base) $1<\beta<\alpha+1$, the so-called {\em univoque set} of numbers which have a…
A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite…
Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of…
This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on…
We study periodic representations in number systems with an algebraic base $\beta$ (not a rational integer). We show that if $\beta$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such…
Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$…
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties…
This paper presents two algorithms on certain computations about Pisot numbers. Firstly, we develop an algorithm that finds a Pisot number $\alpha$ such that $\Q[\alpha] = \F$ given a real Galois extension $\F$ of $\Q$ by its integral…
We survey arithmetic and asymptotic properties of the alternating sum-of-divisors function $\beta$ defined by $\beta(p^a)=p^a-p^{a-1}+p^{a-2}-...+(-1)^a$ for every prime power $p^a$ ($a\ge 1$), and extended by multiplicativity. Certain open…
The vertices of the four dimensional $120$-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group $H_{4}$. There are two special coordinate representations of this root system in which they and…
For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness…
This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier…
We introduce a family of dynamical systems that generate negative $\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of…
Shapovalov elements $\theta_{\beta,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $\beta$ and a positive integer $m$. They relate the canonical…
Recently Sarah Bockting-Conrad introduced the double lowering operator $\psi$ for a tridiagonal pair. Motivated by $\psi$ we consider the following problem about polynomials. Let $\mathbb F$ denote an algebraically closed field. Let $x$…
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''…