Hausdorff dimension of some exceptional sets in Lüroth expansions

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