The efficiency of approximating real numbers by Lüroth expansion

Cao, Chun-Yun; Wu, Jun; Zhang, Zhenliang

doi:10.1007/s10587-013-0033-1

Cited by 18 publications

(4 citation statements)

References 13 publications

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“…Conversely, it is shown in [6] that any sequence of integers with for each must be the Lüroth expansion of some . The Lüroth expansion has been studied extensively in the representation theory of real numbers, probability theory and dynamical systems (see [1, 2, 5, 7] and the monograph of Dajani and Kraaikamp [3]).…”

Section: Introductionmentioning

confidence: 99%

On Lüroth Expansions in Which the Largest Digit Grows With Slowly Increasing Speed

Zhang

Wang²

2022

Bull. Aust. Math. Soc.

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Let $0\leq \alpha \leq \infty $ , $0\leq a\leq b\leq \infty $ and $\psi $ be a positive function defined on $(0,\infty )$ . This paper is concerned with the growth of $L_{n}(x)$ , the largest digit of the first n terms in the Lüroth expansion of $x\in (0,1]$ . Under some suitable assumptions on the function $\psi $ , we completely determine the Hausdorff dimensions of the sets $$\begin{align*}E_\psi(\alpha)=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\} \end{align*}$$ and $$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$

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Section: Introductionmentioning

confidence: 99%

On Lüroth Expansions in Which the Largest Digit Grows With Slowly Increasing Speed

Zhang

Wang²

2022

Bull. Aust. Math. Soc.

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“…where Lüroth series have been investigated, for instance, in [1], [3], [4], [5], [11], [12], [13], [14], [15]. The papers [6], [7] consider a variant of Lüroth series, the socalled alternating Lüroth series .…”

Section: Introductionmentioning

confidence: 99%

Convergence Results forr-Iterated Means of the Denominators of the Lüroth Series

Giuliano

2016

Uniform Distribution Theory

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“…In view of this, it is natural to ask what the Hausdorff dimension of J * (ψ) is? Taking ψ(q) = q −τ (τ > 0), Cao, Wu and Zhang [5] showed that dim H J * (ψ) = 1 τ + 1 .…”

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confidence: 99%