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 \(n\rightarrow \infty \).

中文翻译：

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连续分数的一些出色的Borel–Bernstein定理集

令\（[a_1（x），a_2（x），a_3（x），\ ldots] \）表示实数\（x [in [0,1）\）的连续分数扩展。本文涉及关于（\ {\ {a_n（x）\} _ {n \ geqslant 1} \）的增长率的Borel–Bernstein定理的某些例外集。作为主要结果，集合的Hausdorff维数

$$ \ begin {aligned} E _ {\ sup}（\ psi）= \ left \ {x \ in [0,1）：\ \ limsup \ limits _ {n \ rightarrow \ infty} \ frac {\ log a_n（ x）} {\ psi（n）} = 1 \ right \} \ end {aligned} $$

确定，其中\（\ psi：{\ mathbb {N}} \ rightarrow {\ mathbb {R}} ^ + \）趋于无穷大，如\（n \ rightarrow \ infty \）。