Hausdorff dimension of continued fractions with fast growing partial quotients
Abstract
Let $x\in [0,1)$ be a real number with continued fraction expansion $[a_1(x), a_2(x),a_3(x),\dots ]$, and let ${p_n(x)}/{q_n(x)}$ denote its $n$th convergent. For a real number $\alpha \gt 0$ and a function $\psi : \mathbb {N} \to \mathbb {R}^+$ satisfying $\liminf _{n \to \infty } \psi (n) \gt 0$, we study the sets \begin{align*} E_\psi (\alpha ):=&\{x\in [0,1): a_{n+1}(x)\geq q_n^{\alpha } (x) \psi (n)\text{ for infinitely many }n\},\\ \widetilde {E}_\psi (\alpha ):=&\{x\in [0,1): a_{n+1}(x) \geq q_n^\alpha (x) \psi (n)\text { for all sufficiently large }n\}. \end{align*} We obtain exact formulae for the Hausdorff dimensions of these sets in terms of the exponential and double exponential growth rates of $\psi $. For $E_\psi (\alpha )$, the Hausdorff dimension is determined by the solution of a pressure equation associated with the Gauss map, exhibiting a continuous transition between the classical results of Good (1941) and Sun and Wu (2014). For $\widetilde {E}_\psi (\alpha )$, the Hausdorff dimension depends on the upper double exponential growth rate of $\psi $ and admits a simple explicit expression.
