On the fast Khintchine spectrum in continued fractions

Abstract

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_n(x), n\ge 1\}$. Let $\psi : \mathbb{N } \rightarrow \mathbb{N }$ be a function with $\psi (n)/n\rightarrow \infty$ as $n\rightarrow \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set

$$\begin{aligned} E(\psi ):=\left\{ x\in [0,1): \lim _{n\rightarrow \infty }\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)=1\right\} \end{aligned}$$

is completely determined without any extra condition on $\psi$. This fills a gap of the former work in Fan et al. (Ergod Theor Dyn Syst 29:73–109, 2009).