Subexponentially increasing sums of partial quotients in continued fraction expansions

TL;DR

This paper analyzes the Hausdorff dimension of sets of irrationals with partial quotient sums growing at subexponential rates, revealing a dimension jump at a critical exponential threshold, using multifractal analysis.

Contribution

It determines the Hausdorff dimension of sets where partial quotient sums grow subexponentially, especially in the critical case xp(n^{1/2}), showing a dimension jump from 1 to 1/2.

Findings

Hausdorff dimension of E_ is 1/2 for  in [1/2, 1)

Dimension jumps from 1 to 1/2 at (n)=xp(n^{1/2})

Results extend understanding of growth rates of partial quotients in continued fractions.

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 $\varphi: \mathbb{N} \rightarrow \mathbb{N}$, one is interested in the Hausdorff dimension of the sets\[E\_\varphi = \left\{x\in (0,1): \lim\_{n\to\infty} \frac {S\_n(x)} {\varphi(n)} =1\right\}.\]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 $\gamma \in [1/2, 1)$, $E\_\varphi$ has Hausdorff dimension $1/2$. Thus, surprisingly, the dimension has a jump from $1$ to $1/2$ at $\varphi(n)=\exp(n^{1/2})$. In a similar way, the distribution of the largest partial quotient is also studied.