On the exceptional sets concerning the leading partial quotient in continued fractions

Zhang, Mengjie; Ma, Chao

doi:10.1016/j.jmaa.2021.125110

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2022

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“…Theorem 1.3 also implies the following result. For more results concerning the largest digits in Lüroth expansions and continued fraction expansions, see [10,[15][16][17]. For the definitions and elementary properties of Hausdorff dimension, Falconer's book [4] is recommended.…”

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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