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 each \(n \in \mathbb {N}\) , define \(T_n(x):=\max \{a_1(x),\dots ,a_n(x)\}\) . We study the Hausdorff dimension of the set \(\begin{aligned} E(\psi ):=\left\{ x \in (0,1)\backslash \mathbb {Q}: \lim _{n \rightarrow \infty } \frac{T_n(x)}{\psi (n)} =1\right\} , \end{aligned}\) where \(\psi : \mathbb {N}\rightarrow \mathbb {R}^+\) is an increasing function satisfying \(\psi (n) \rightarrow \infty \) as \(n\rightarrow \infty \) . Liao and Rams (Math Proc Cambridge Philos Soc 160: 401–412, 2016) observed a discontinuous jump in the Hausdorff dimension of \(E(\psi )\) from 1 to 1/2 in the class \(\psi (n)=\exp (n^r)\) at \(r=1/2\) . In this paper, we provide a detailed analysis of this dimension gap and establish sufficient conditions for \(E(\psi )\) to have Hausdorff dimension 1 or 1/2. These results generalize the findings of Liao and Rams (Math Proc Cambridge Philos Soc 160: 401–412, 2016), and Fang and Liu (Fractals 29: 2150099, 2021). Moreover, we give a negative answer to a question posed by Fang and Liu (Fractals 29: 2150099, 2021).
