On upper and lower fast Knintchine spectra of continued fractions

TL;DR

This paper analyzes the multifractal properties of continued fractions by determining the Hausdorff dimensions of sets where the growth rate of the sum of logs of partial quotients, scaled by a rapidly increasing function, reaches a specific limit.

Contribution

It provides exact formulas for the upper and lower fast Khintchine spectra, extending previous results by Liao and Rams (2016).

Findings

Explicit formulas for the spectra are derived.

The results strengthen previous partial findings.

The spectra are characterized by Hausdorff dimensions.

Abstract

Let $\psi:\mathbb{N}\to \mathbb{R}^+$ be a function satisfying $\phi(n)/n\to \infty$ as $n \to \infty$. We investigate from a multifractal analysis point of view the growth rate of the sums $\sum^n_{k=1}\log a_k(x)$ relative to $\psi(n)$, where $[a_1(x),a_2(x), a_3(x)\cdots]$ denotes the continued fraction expansion of $x\in (0,1)$. The upper (resp. lower) fast Khintchine spectrum is defined by the Hausdorff dimension of the set of all points $x$ for which the upper (resp. lower) limit of $\frac{1}{\psi(n)}\sum^n_{k=1}\log a_k(x)$ is $1$. The precise formulas of these two spectra are completely determined, which strengthens a result of Liao and Rams (2016).