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ON LÜROTH EXPANSIONS IN WHICH THE LARGEST DIGIT GROWS WITH SLOWLY INCREASING SPEED
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 107 / Issue 2 / April 2023
- Published online by Cambridge University Press:
- 23 June 2022, pp. 204-214
- Print publication:
- April 2023
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- Article
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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
$$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$
