Subexponentially increasing sums of partial quotients in continued fraction expansions

Abstract

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S\_n(x)=\sum \_{j=1}^{n}a\_j(x)$, where $x=[a\_1(x),a\_2(x),\cdots ]$ is the continued fraction expansion of an irrational $x\in (0,1)$. Precisely, for an increasing function $\phi :\mathbb{N}\to \mathbb{N}$, one is interested in the Hausdorff dimension of the sets\[E\_φ= \left\{x\in (0,1): \lim\_{n\to\infty} \frac {S\_n(x)} {φ(n)} =1\right\}.\]Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\exp (n^{\gamma }),\gamma \in [1/2,1)$. We show that when $\gamma \in [1/2,1)$, $E\_\phi$ has Hausdorff dimension $1/2$. Thus, surprisingly, the dimension has a jump from $1$ to $1/2$ at $\phi (n)=\exp (n^{1/2})$. In a similar way, the distribution of the largest partial quotient is also studied.

12 pages. More details for the proof of Theorem 1.2. are added

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