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 $α$-Rosen fractions. The metrical properties of these $α$-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are closely related to similar region for the 'classical' Rosen fraction. This allows us to unify and generalize results of diophantine approximation from the literature.
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 consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.
Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the $n$th partial quotient of the continued fraction expansion of $x$ in the field of formal Laurent series. We consider the sets of $x$ such that $°A_{n+1}(x)+\cdots+°A_{n+k}(x)~\ge~Φ(n)$ holds for infinitely many $n$ and for all $n$ respectively, where $k\ge1$ is an integer and $Φ(n)$ is a positive function defined on $\mathbb{N}$. We determine the size of these…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the continued fraction transformations associated with expansions.
The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecut…
The purpose of this paper is to describe the relation between the Legendre and the Lenstra constants. Indeed we show that they are equal whenever the Legendre constant exists; in particular, this holds for both Rosen continued fractions and $α$-continued fractions. We also give the explicit value of the entropy of the Rosen map with respect to the absolutely continuous invariant probability measure.
We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.
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. It follows that the underlying maps have equal entropy.
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 approximations for the alpha for which the Legendre constant is larger than the Hurwitz constant.
We describe how to represent Rosen continued fractions by paths in a class of graphs that arise naturally in hyperbolic geometry. This representation gives insight into Rosen's original work about words in Hecke groups, and it also helps us to identify Rosen continued fraction expansions of shortest length.
In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.
A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.
In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.
We introduce and study in detail a special class of backward continued fractions that represents a generalization of Rényi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we construct the natural extension for the transformation that generates the Rényi-type expansion. Also we define the associated random system with complete connections whose ergodic behavior allows us to prove a variant of Gauss-Kuzmin-type theorem.
The nearest integer continued fraction of a real number $x$ from $[-1/2, 1/2)$ is defined. Some metrical properties of these expansions are presented. We define the approximation coefficients and give an important result on them. The main result consists in obtaining a stationary state for the transformation $τ_{1/2}$ which is absolutely continuous with respect to the Lebesgue measure.
We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the α-continued fractions, for each $α$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced α-variants.
In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes …
We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions…
We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Per…
