Hausdorff dimension of some exceptional sets in L\"{u}roth expansions

Abstract

In this paper, we study the metrical theory of the growth rate of digits in L\"{u}roth expansions. More precisely, for $ x\in \left( 0,1 \right] $, let $ \left[ d_1\left( x \right) ,d_2\left( x \right) ,\cdots \right] $ denote the L\"{u}roth expansion of $ x $, we completely determine the Hausdorff dimension of the following sets \begin{align*} E_{\mathrm{sup}}\left( \psi \right) =\Big\{ x\in \left( 0,1 \right] :\limsup\limits_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{\psi \left( n \right)}=1 \Big\} , \end{align*} \begin{align*} E\left( \psi \right) =\Big\{ x\in \left( 0,1 \right] :\lim_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{\psi \left( n \right)}=1 \Big\} \end{align*} and \begin{align*} E_{\mathrm{inf}}\left( \psi \right) =\Big\{ x\in \left( 0,1 \right] : \liminf_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{\psi \left( n \right)}=1 \Big\} , \end{align*} where $ \psi :\mathbb{N} \rightarrow \mathbb{R} ^+ $ is an arbitrary function satisfying $ \psi \left( n \right) \rightarrow \infty$ as $n\rightarrow \infty$.