Some exceptional sets of Borel–Bernstein theorem in continued fractions

Abstract

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 $n\rightarrow \infty$.