On upper and lower fast Knintchine spectra of continued fractions

Abstract

Let $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$ be a function satisfying $\varphi (n)/n\to \infty$ as $n\to \infty$. We investigate from a multifractal analysis point of view the growth rate of the sums ${\sum }_{k=1}^n\log a_k(x)$ 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 $\frac{1}{\psi (n)}{\sum }_{k=1}^n\log a_k(x)$ is $1$. The precise formulas of these two spectra are completely determined, which strengthens a result of Liao and Rams (2016).

References in corpus (2)

• On the fast Khintchine spectrum in continued fractions
• Spectrum of weighted Birkhoff average