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 $$\begin{aligned} E_{\sup }(\psi )=\left\{ x\in [0,1):\ \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} \end{aligned}$$ is determined, where $$\psi :{\mathbb {N}}\rightarrow {\mathbb {R}}^+$$ tends to infinity as $$n\rightarrow \infty $$ .