scholarly journals HAUSDORFF DIMENSION FOR THE SET OF POINTS CONNECTED WITH THE GENERALIZED JARNÍK–BESICOVITCH SET

2020 ◽  
pp. 1-29
Author(s):  
AYREENA BAKHTAWAR
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Gauss Map ◽  
Continuous Functions ◽  
Partial Quotient ◽  
Continued Fraction Expansion ◽  
Set Of Points

Abstract In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$ , $$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$ holds for infinitely many $n\in \mathbb {N}$ , where h and $\tau $ are positive continuous functions, T is the Gauss map and $a_{n}(x)$ denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of $r,\tau (x)$ and $h(x)$ we obtain various classical results including the famous Jarník–Besicovitch theorem.

On the exceptional sets in Sylvester continued fraction expansion

2015 ◽  
Vol 11 (08) ◽  
pp. 2369-2380
Author(s):  
Zhen-Liang Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Exceptional Sets ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Set Of Points

In this paper, we study some exceptional sets of points whose partial quotients in their Sylvester continued fraction expansions obey some restrictions. More precisely, for α ≥ 1 we prove that the Hausdorff dimension of the set [Formula: see text] is one. In addition, we find that the points whose partial quotients in their Sylvester continued fraction expansions obey some property of divisibility have the same Engel continued fraction expansion and Sylvester continued fraction expansion. And we establish that the set of points whose Engel continued fraction expansion and Sylvester continued fraction expansion coincide is uncountable.

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.

The distribution of the largest digit in continued fraction expansions

2009 ◽  
Vol 146 (1) ◽  
pp. 207-212 ◽  
Author(s):  
JUN WU ◽  
JIAN XU
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Continued Fraction Expansions

AbstractLet [a1(x), a2(x), . . .] be the continued fraction expansion of x ∈ [0,1). Write Tn(x)=max{ak(x):1 ≤ k ≤ n}. Philipp [6] proved that Okano [5] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T(x)=1/log k. In this paper we show that, for any α ≥ 0, the set is of Hausdorff dimension 1.

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.

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].

scholarly journals A Specialised Continued Fraction

1993 ◽  
Vol 45 (5) ◽  
pp. 1067-1079 ◽  
Author(s):  
A. J. Van Der Poorten ◽  
J. Shallit
Keyword(s):  
Continued Fraction ◽  
Formal Series ◽  
Rational Integer ◽  
Partial Quotient ◽  
Continued Fraction Expansion ◽  
Integer Coefficients ◽  
Positive Degree ◽  
Partial Quotients ◽  
Special Case

AbstractWe display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ΣchX-h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple arguments, let alone examples, demonstrate that it is noteworthy if those partial quotients happen to have rational integer coefficients only. In that special case one may replace the variable X by an integer ≥ 2; that is: one may 'specialise' and thereby proceed to obtain the regular continued fraction expansion of values of the series. And that is significant because, generally, it is difficult to obtain the explicit continued fraction expansion of a number presented in different shape. Our example leads to a series with a specialisable continued fraction expansion and, a little surprisingly, our arguments suggest that the phenomenon of specialisability for series of the kind appearing here may be reserved to just the special subclass of series we happen to have stumbled upon.

scholarly journals On Khintchine exponents and Lyapunov exponents of continued fractions

2009 ◽  
Vol 29 (1) ◽  
pp. 73-109 ◽  
Author(s):  
AI-HUA FAN ◽  
LING-MIN LIAO ◽  
BAO-WEI WANG ◽  
JUN WU
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Continued Fractions ◽  
Multifractal Analysis ◽  
Continued Fraction ◽  
Constant Function ◽  
Point Of View ◽  
Lyapunov Spectrum ◽  
Continued Fraction Expansion

AbstractAssume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by $\gamma (x):=\lim _{n\to \infty }({1}/{n}) \sum _{j=1}^n \log a_j(x)$ when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of $\xi \in [0, +\infty )$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma ^{\varphi }(x):=\lim _{n\to \infty }({1}/({\varphi (n)}))\sum _{j=1}^n \log a_j(x)$ are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

scholarly journals Subexponentially increasing sums of partial quotients in continued fraction expansions

2015 ◽  
Vol 160 (3) ◽  
pp. 401-412 ◽  
Author(s):  
LINGMIN LIAO ◽  
MICHAŁ RAMS
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Point Of View ◽  
Partial Quotient ◽  
Formula One ◽  
Formula Group ◽  
Partial Quotients ◽  
Image Position ◽  
Continued Fraction Expansions ◽  
Increasing Function

AbstractWe investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S_{n}(x)=\sum_{j=1}^n a_{j}(x)$, where x = [a1(x), a2(x), . . .] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : $\mathbb{N}$ → $\mathbb{N}$, one is interested in the Hausdorff dimension of the set E_\varphi = \left\{x\in (0,1): \lim_{n\to\infty} \frac {S_n(x)} {\varphi(n)} =1\right\}. Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(nγ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), Eϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ(n) = exp(n1/2). In a similar way, the distribution of the largest partial quotient is also studied.

scholarly journals The bifurcation locus for numbers of bounded type

2021 ◽  
pp. 1-31
Author(s):  
CARLO CARMINATI ◽  
GIULIO TIOZZO
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Measure Zero ◽  
Period Doubling ◽  
Unit Interval ◽  
Continued Fraction Expansion ◽  
Quadratic Polynomials ◽  
The Family ◽  
Real Quadratic ◽  
Period Doubling Bifurcations

Abstract We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$ , and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$ . The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$ .

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