On the growth rate of partial quotients in Engel continued fractions

Let $(b_n(x))_{n\geq1}$ be the sequence of the partial quotients of the Engel continued fraction expansion of an irrational number $x\in(0,1)$. This paper is concerned with the Hausdorff dimension of some exceptional sets related to the growth rate of $(b_n(x))_{n\geq1}$. As a main result, we obtain the Hausdorff dimension of the set \[ E_{\inf}(\psi)=\left\{x\in[0,1): \liminf\limits_{n\to\infty}\frac{\log b_n(x)}{\psi(n)}=1\right\}, \] where $\psi:\mathbb{N}\rightarrow\mathbb{R}^+$ is a function satisfying $\psi(n) \to \infty$ as $n\to\infty$.