Some exceptional sets of Borel–Bernstein theorem in continued fractions

Let $[a_1(x),a_2(x), a_3(x),\ldots ] denote the continued fraction expansion of a real number x \in [0,1) . This paper is concerned with certain exceptional sets of the Borel–Bernstein Theorem on the growth rate of \{a_n(x)\}_{n\geqslant 1} . As a main result, the Hausdorff dimension of the set \begin{aligned} E_{\sup }(\psi )=\left\{ x\in [0,1):\ \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\}...