On the Largest Partial Quotients in Continued Fraction Expansions

Fractals ◽  
2020 ◽  
Author(s):  
Lulu Fang ◽  
Jian Liu
Keyword(s):  
Continued Fraction ◽  
Partial Quotients ◽  
Continued Fraction Expansions

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.

Slow increasing functions and the largest partial quotients in continued fraction expansions

2016 ◽  
Vol 164 (1) ◽  
pp. 1-14 ◽  
Author(s):  
JINHUA CHANG ◽  
HAIBO CHEN
Keyword(s):  
Level Sets ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Positive Function ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Increasing Functions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

scholarly journals On the frequency of partial quotients of regular continued fractions

2009 ◽  
Vol 148 (1) ◽  
pp. 179-192 ◽  
Author(s):  
AI-HUA FAN ◽  
LINGMIN LIAO ◽  
JI-HUA MA
Keyword(s):  
Variational Principle ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.

ON THE LARGEST DEGREE OF THE PARTIAL QUOTIENTS IN CONTINUED FRACTION EXPANSIONS OVER THE FIELD OF FORMAL LAURENT SERIES

2013 ◽  
Vol 09 (05) ◽  
pp. 1237-1247 ◽  
Author(s):  
LUMING SHEN ◽  
JIAN XU ◽  
HUIPING JING
Keyword(s):  
Continued Fraction ◽  
Laurent Series ◽  
Type Theorem ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Iterated Logarithm ◽  
Partial Quotients ◽  
Almost All ◽  
Continued Fraction Expansions

For x ∈ I, let [A1(x), A2(x), …] be the continued fraction expansions over the field of Laurent series, write Ln(x) ≔ max { deg A1(x), deg A2(x), …, deg An(x)}, which is called the largest degree of partial quotients. In this paper, we give an iterated logarithm type theorem for Ln(x), and by which, we get that for P-almost all x ∈ I, [Formula: see text]. Also the Hausdorff dimensions of the related exceptional sets are determined.

scholarly journals A Transcendental Unbounded Continued Fraction Expansions over A Finite Field

2021 ◽  
Vol 27 (1) ◽  
pp. 115-122
Author(s):  
Rima Ghorbel ◽  
Hassen Kthiri
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Formal Power

Let Fq be a finite field and Fq((X−1 )) the field of formal power series with coefficients in Fq. The purpose of this paper is to exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients

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.

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