Summary: For an irrational number \(x \in [0, 1)\), let \(x = [a_1(x), a_2(x), \ldots]\) be its continued fraction expansion and \(\{\frac{p_n(x)}{q_n(x)}, n \geq 1 \}\) be the sequence of rational convergents. Then, for any \(\alpha, \beta \in [0, + \infty]\) with \(\alpha \leq \beta\), the Hausdorff dimension of the following set \[F(\alpha, \beta) = \left\{x \in [0, 1) : \underset{n \rightarrow \infty}{\liminf} \frac{\log a_{n + 1}(x)}{\log q_n(x)} = \alpha, \underset{n \rightarrow \infty}{\limsup} \frac{\log a_{n + 1}(x)}{\log q_n(x)} = \beta \right\}\] admits a dichotomy: it is either \(\frac{1}{\beta + 2}\) or \(\frac{2}{\beta + 2}\) according as \(\alpha > 0\) or not.