Hausdorff dimension of certain sets arising in 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.

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Subexponentially increasing sums of partial quotients in continued fraction expansions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000742 ◽

2015 ◽

Vol 160 (3) ◽

pp. 401-412 ◽

Cited By ~ 9

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.

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On the exceptional sets in Sylvester continued fraction expansion

International Journal of Number Theory ◽

10.1142/s1793042115501092 ◽

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.

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

Mathematics ◽

10.3390/math9030255 ◽

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.

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ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

Stochastics and Dynamics ◽

10.1142/s0219493704000900 ◽

2004 ◽

Vol 04 (01) ◽

pp. 63-76 ◽

Cited By ~ 24

Author(s):

OLIVER JENKINSON

Keyword(s):

Hausdorff Dimension ◽

Finite Subset ◽

Continued Fraction ◽

Cantor Set ◽

Computational Method ◽

Accurate Approximation ◽

Natural Numbers ◽

Important Special Case ◽

The One ◽

Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

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