Some exceptional sets of Borel–Bernstein theorem in continued fractions

Let [a1(x),a2(x),a3(x),…]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[a_1(x),a_2(x), a_3(x),\ldots ]$$\end{document} denote the continued fraction expansion of a real number x∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in [0,1)$$\end{document}. This paper is concerned with certain exceptional sets of the Borel–Bernstein Theorem on the growth rate of {an(x)}n⩾1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_n(x)\}_{n\geqslant 1}$$\end{document}. As a main result, the Hausdorff dimension of the set Esup(ψ)=x∈[0,1):lim supn→∞logan(x)ψ(n)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} E_{\sup }(\psi )=\left\{ x\in [0,1):\ \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} \end{aligned}$$\end{document}is determined, where ψ:N→R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi :{\mathbb {N}}\rightarrow {\mathbb {R}}^+$$\end{document} tends to infinity as n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}.