Let \(x = [0;a_1(x),a_2(x),\ldots]\) denote the classical continued fraction expansion of the irrational number \(x \in [0,1)\). In this article, the authors consider the size of the sets \[ E(\alpha,\beta) := \left\{x \in [0,1]\setminus \mathbb{Q}: \liminf_{n \to \infty} \frac{\log a_n(x)}{\log n} = \alpha, \limsup_{n \to \infty}\frac{\log a_n(x)}{\log n} = \beta \right\}, \] where \(0 \leq \alpha \leq \beta \leq \infty\), both from the point of view of Baire classification and Hausdorff dimension. Clearly, analyzing the almost sure behaviour with respect to Lebesgue measure yields that \(E(0,1)\) is of full measure and all other sets have measure zero.
In Theorem 1.1, which treats the Baire classification, it is shown that the set \(E(\alpha, \beta)\) is residual if and only if \((\alpha,\beta) = (0,\infty)\).
In Theorem 1.2, which treats the Hausdorff dimension, the authors show that the Hausdorff dimension of \(E(\alpha,\beta)\) is independent of \(\beta\). Furthermore, for all \(\alpha,\beta \in [0,\infty]\) with \(\alpha \leq \beta\), \[ \dim_{H} E(\alpha,\beta) = \begin{cases} 1, &\text{ if } \alpha = 0,\\ \frac{1}{2}, &\text{ if } \alpha > 0. \end{cases} \] Theorem 1.2 can be compared to the recent result of [L. Fang et al., Ramanujan J. 56, No. 3, 891–909 (2021; Zbl 1481.11078)] who proved that for any \(\alpha > 0\), \(\dim_H\left(\bigcup\limits_{\beta \in [0,\infty]} E(\alpha,\beta)\right) = \frac{1}{2}\) and for any \(\beta > 0\), \(\dim_H\left(\bigcup\limits_{\alpha \in [0,\infty]} E(\alpha,\beta)\right) = 1\).
From a naive point of view, the combination of Theorems 1.1 and 1.2 shows an interesting counterintuitive behaviour: \(E(0,\infty)\) is considered “large” with respect to the Baire classification (as complement of a first category set), but has Hausdorff dimension \(0\). On the other hand, \(E(\alpha,\beta)\) for \(\alpha > 0\) has positive Hausdorff dimension, but is of first category and thus is a “small set” with respect to the Baire classification.
As another result, the authors consider the set of all \(x\) such that the sequence of its partial quotients is non-decreasing, i.e., let \[ \Lambda := \left\{x \in [0,1]\setminus \mathbb{Q}: a_n(x) \leq a_{n+1}(x) \;\;\text{ for all } n \geq 1 \right\}. \] It was shown in [L. Fang et al., Acta Math. Sci., Ser. B, Engl. Ed. 41, No. 6, 1896–1910 (2021; Zbl 1513.11143)] that \[ \dim_H\left(\Lambda \cap \left(\bigcup\limits_{\beta \in [0,\infty]} E(\alpha,\beta)\right)\right) = \begin{cases} 0 &\text{ if } 0 \leq \alpha \leq 1, \\ \frac{\alpha-1}{2\alpha} &\text{ if } \alpha > 1. \end{cases} \] In Theorem 1.3, it is established that the same Hausdorff estimates hold for any fixed \(\beta \geq \alpha\), i.e., for any \(0 \leq \alpha \leq \beta \leq \infty\), we have \[ \dim_H\left(\lambda \cap E(\alpha,\beta)\right) = \begin{cases} 0 &\text{ if } 0 \leq \alpha \leq 1, \\ \frac{\alpha-1}{2\alpha} &\text{ if } \alpha > 1. \end{cases} \] The proofs of all three theorems are elementary and very short (thus the paper is only 7 pages long), but are nevertheless well readable.