Upper and lower fast Khintchine spectra in continued fractions

Abstract

For an irrational number $x\in [0,1)$, let $x=[a\_1(x),a\_2(x),\cdots ]$ be its continued fraction expansion. Let $\psi :\mathbb{N}\to \mathbb{N}$ be a function with $\psi (n)/n\to \infty$ as $n\to \infty$. The (upper, lower) fast Khintchine spectrum for $\psi$ is defined as the Hausdorff dimension of the set of numbers $x\in (0,1)$ for which the (upper, lower) limit of $\frac{1}{\psi (n)}\sum \_{j=1}^n\log a\_j(x)$ is equal to $1$. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.

13 pages. Motivation and details of proofs are added

References in corpus (1)

• Subexponentially increasing sums of partial quotients in continued fraction expansions

Cited by in corpus (6)

• Multifractal analysis of the convergence exponent in continued fractions
• Limit theorems for counting large continued fraction digits
• Some exceptional sets of Borel-Bernstein Theorem in continued fractions
• On upper and lower fast Knintchine spectra of continued fractions
• On the exponent of convergence of Engel series
• Full dimensional sets of reals whose sums of partial quotients increase in certain speed