scholarly journals The relative growth rate of the largest partial quotient to the sum of partial quotients in continued fraction expansions

2016 ◽  
Vol 163 ◽  
pp. 482-492
Author(s):  
Zhenliang Zhang ◽  
Meiying Lü
Keyword(s):  
Growth Rate ◽  
Relative Growth Rate ◽  
Continued Fraction ◽  
Partial Quotient ◽  
Relative Growth ◽  
Partial Quotients ◽  
Continued Fraction Expansions

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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