scholarly journals A remark on Liao and Rams' result on the distribution of the leading partial quotient with growing speeden1/2in continued fractions

2017 ◽  
Vol 355 (7) ◽  
pp. 734-737
Author(s):  
Liangang Ma
Keyword(s):  
Continued Fractions ◽  
Partial Quotient

Metrical property for GCFϵ expansion with the parameter function ϵ(k) = c(k + 1)

2015 ◽  
Vol 11 (07) ◽  
pp. 2065-2072 ◽  
Author(s):  
Ting Zhong ◽  
Quanwu Mu ◽  
Luming Shen
Keyword(s):  
Continued Fractions ◽  
Partial Quotient ◽  
The Real ◽  
Metric Properties ◽  
Metrical Property ◽  
Parameter Function ◽  
Generalized Continued Fractions

This paper is concerned with the metric properties of the generalized continued fractions (GCFϵ) with the parameter function ϵ(kn), where kn is the nth partial quotient of the GCFϵ expansion. When -1 < ϵ(kn) ≤ 1, Zhong [Metrical properties for a class of continued fractions with increasing digits, J. Number Theory128 (2008) 1506–1515] obtained the following metrical properties: [Formula: see text] which are entirely unrelated to the choice of ϵ(kn) ∈ (-1, 1]. Here we deal with the case of ϵ(k) = c(k + 1) with constant c ∈ (0, ∞). It is proved that: [Formula: see text] which change with the real c ∈ (0, ∞). Note that [Formula: see text] as c → 0, it indicates that when c → 0, the GCFϵ has the same metrical property as the case of -1 < ϵ(kn) ≤ 1.

A generalization of the Jarník–Besicovitch theorem by continued fractions

2015 ◽  
Vol 36 (4) ◽  
pp. 1278-1306 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Gauss Map ◽  
Partial Quotient ◽  
Image Position ◽  
Positive Functions ◽  
Besicovitch Sets

We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where $T$ is the Gauss map and $a_{n}(x)$ is the $n$th partial quotient of $x$. In this paper, we consider the size of the generalized Jarník–Besicovitch set $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where ${\it\tau}(x)$ and $f(x)$ are positive functions defined on $[0,1]$.

scholarly journals Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction

2019 ◽  
Vol 11 (1) ◽  
pp. 33-41 ◽  
Author(s):  
I.B. Bilanyk ◽  
D.I. Bodnar ◽  
L. Buyak
Keyword(s):  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Linear Recurrence ◽  
Partial Quotient ◽  
Term Linear ◽  
Recurrent Relations ◽  
Multiple Power ◽  
Linear Recurrence Relation ◽  
Branched Continued Fraction

The quotient of two linearly independent solutions of a four-term linear recurrence relation is represented in the form of a branched continued fraction with two branches of branching by analogous with continued fractions. Formulas of partial numerators and partial denominators of this branched continued fraction are obtained. The solutions of the recurrence relation are canonic numerators and canonic denominators of $\mathcal{B}$-figured approximants. Two types of figured approximants $\mathcal{A}$-figured and $\mathcal{B}$-figured are often used. A $n$th $\mathcal{A}$-figured approximant of the branched continued fraction is obtained by adding a next partial quotient to the $(n-1)$th $\mathcal{A}$-figured approximant. A $n$th $\mathcal{B}$-figured approximant of the branched continued fraction is a branched continued fraction that is a part of it and contains all those elements that have a sum of indexes less than or equal to $n$. $\mathcal{A}$-figured approximants are widely used in proving of formulas of canonical numerators and canonical denominators in a form of a determinant, $\mathcal{B}$-figured approximants are used in solving the problem of corresponding between multiple power series and branched continued fractions. A branched continued fraction of the general form cannot be transformed into a constructed branched continued fraction. For calculating canonical numerators and canonical denominators of a branched continued fraction with $N$ branches of branching, $N>1$, the linear recurrent relations do not hold. $\mathcal{B}$-figured convergence of the constructed fraction in a case when coefficients of the recurrence relation are real positive numbers is investigated.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

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