Related papers: On the Lenstra constant associated to the Rosen co…
The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the $\alpha$-Rosen fractions. The metrical…
In this article we generalize Borel's classical approximation results for the regular continued fraction expansion to the alpha-Rosen fraction expansion, using a geometric method. We give a Haas-Series-type result about all possible good…
The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate…
The present paper studies continuity of generalized entropy functions and relative entropies defined using the notion of a deformed logarithmic function. In particular, two distinct definitions of relative entropy are discussed. As an…
We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued…
This is a brief historical note about famous Legendre's criterium for convergent of continued fraction expansion. The paper is written in Russian.
This document expands upon the relationship between discrete and continuous entropy given in (Phys. Rev. Lett. 110 130407), \Violating Continuous Variable Einstein-Podolsky-Rosen Steering with Discrete Measurements". We provide a detailed…
We give natural extensions for the alpha-Rosen continued fractions of Dajani et al. for a set of small alpha values by appropriately adding and deleting rectangles from the region of the natural extension for the standard Rosen fractions.…
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and,…
In this paper, we investigate the monotone property of the continued fractions $G(m,\lambda)$ as a function of $m$ and $\lambda$. In particular, we obtain new inequality for the relative continued fractions.
We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of…
We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural…
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are…
As a natural counterpart to Nakada's $\alpha$-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter…
We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.
We study the ergodic theory of a one-parameter family of interval maps T_alpha arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of T_alpha to be Hoelder-continuous in the…
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood…
A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…
The value of the alpha constant, known to be equal to an algebraic expression in terms of pi and entire numbers related to certain group volumes, is derived from the relativistic structure group of a geometric unified theory, its subgroups…