Let \([a_1(x),a_2(x),a_3(x),\ldots ]\) denote the continued fraction expansion of an irrational number \(x\in [0,1)\) . For any function \(\psi :\mathbb {N}\rightarrow \mathbb {R}^+\) satisfying \(\psi (n)\rightarrow \infty \) as \(n\rightarrow \infty \) , define \(\begin{aligned} E_{\sup }(\psi ):=\left\{ x\in [0,1)\backslash \mathbb {Q}:\, \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} . \end{aligned}\) In (Ramanujan J 56:891–909, 2021), the authors determined the Hausdorff dimension of \(E_{\sup }(\psi )\) under the condition that the limit \(\lim _{n\rightarrow \infty }\psi (n)/n\) exists. In this paper, we remove this hypothesis and provide a complete description of the Hausdorff dimension of \(E_{\sup }(\psi )\) .
