ON THE INCREASING PARTIAL QUOTIENTS OF CONTINUED FRACTIONS OF POINTS IN THE PLANE

2021 ◽  
pp. 1-8
Author(s):  
MEIYING LÜ ◽  
ZHENLIANG ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Large N ◽  
Partial Quotients ◽  
Positive Integers

Abstract For any x in $[0,1)$ , let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $ . For any positive integers s and t, we study the set $$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$ and determine its Hausdorff dimension.

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

2021 ◽  
pp. 1-29
Author(s):  
LINGLING HUANG ◽  
CHAO MA
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Natural Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Nonnegative Number ◽  
Partial Quotients

Abstract This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

scholarly journals Limits of unbounded sequences of continued fractions

1990 ◽  
Vol 41 (3) ◽  
pp. 509-512
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Quotients ◽  
Limit Points ◽  
Necessary And Sufficient ◽  
Positive Integers

Let X = {xk}k≥1 be a sequence of positive integers. Let Qk = [O;xk,xk−1,…,x1] be the finite continued fraction with partial quotients xi(1 ≤ i ≤ k). Denote the set of the limit points of the sequence {Qk}k≥1 by Λ(X). In this note a necessary and sufficient condition is given for Λ(X) to contain no rational numbers other than zero.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

scholarly journals CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

2002 ◽  
Vol 45 (3) ◽  
pp. 653-671 ◽  
Author(s):  
J. L. Davison
Keyword(s):  
Rate Of Convergence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Subject Classification ◽  
Specific Sequence ◽  
Secondary 11B37 ◽  
Quadratic Irrational ◽  
Infinite Word ◽  
Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

THE DISTAL AND ASYMPTOTIC SETS FOR CONTINUED FRACTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950139
Author(s):  
WEIBIN LIU ◽  
SHUAILING WANG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Underlying Space ◽  
And Topology

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

scholarly journals Good’s theorem for Hurwitz continued fractions

2020 ◽  
Vol 16 (07) ◽  
pp. 1433-1447
Author(s):  
Gerardo Gonzalez Robert
Keyword(s):  
Hausdorff Dimension ◽  
Complex Plane ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Analogous Result ◽  
Complex Numbers ◽  
Real Numbers ◽  
Dimension Formula

Good’s Theorem for regular continued fraction states that the set of real numbers [Formula: see text] such that [Formula: see text] has Hausdorff dimension [Formula: see text]. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [Formula: see text] satisfies [Formula: see text] has Hausdorff dimension [Formula: see text], half of the ambient space’s dimension.

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
Periodic Continued Fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

scholarly journals Non-periodic continued fractions in hyperelliptic function fields

2001 ◽  
Vol 64 (2) ◽  
pp. 331-343 ◽  
Author(s):  
Alfred J. van der Poorten
Keyword(s):  
Exponential Growth ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Function Fields ◽  
Square Root ◽  
90Th Birthday ◽  
Continued Fraction Expansion ◽  
Generic Polynomial ◽  
Partial Quotients ◽  
Hyperelliptic Function

Dedicated to George Szekeres on his 90th birthdayWe discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.

A dimensional result in continued fractions

2014 ◽  
Vol 10 (04) ◽  
pp. 849-857 ◽  
Author(s):  
Yu Sun ◽  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Dimensional Theory ◽  
Cambridge Philos ◽  
Dimension Formula ◽  
Growth Properties

Given x ∈ (0, 1), let [a1(x), a2(x), a3(x),…] be the continued fraction expansion of x and [Formula: see text] be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {an(x), n ≥ 1} and proved that for any β > 0, the set [Formula: see text] is of Hausdorff dimension [Formula: see text]. In this paper, we consider, for any β > 0, the set [Formula: see text] and show that the Hausdorff dimension of F(β) is [Formula: see text].

Some properties of the continued fraction expansion of (m/n) e1/q

1970 ◽  
Vol 67 (1) ◽  
pp. 67-74 ◽  
Author(s):  
K. R. Matthews ◽  
R. F. C. Walters
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Positive Integers

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

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