On the fast Khintchine spectrum in continued fractions

Abstract

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ 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 $\psi(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$ E(\psi):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{\psi(n)}\sum_{j=1}^n\log a_j(x)=1\Big} $$ is completely determined without any extra condition on $\psi$.