Some exceptional sets of Borel-Bernstein Theorem in continued fractions

Abstract: Let $[a_1(x),a_2(x), a_3(x),\cdots]$ 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 ${a_n(x)}{n\geq1}$. As a main result, the Hausdorff dimension of the set [ E{\sup}(\psi)=\left{x\in[0,1):\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{\psi(n)}=1\right} ] is determined, where $\psi:\mathbb{N}\rightarrow\mathbb{R}^+$ tends to infinity as $n\to\infty$.