On the fast Khintchine spectrum in continued fractions

Abstract

For $x\in [0,1)$, let $x=[a_1(x),a_2(x),...]$ be its continued fraction expansion with partial quotients $a_n(x),n\ge 1$. Let $\psi :\mathbb{N}\to \mathbb{N}$ be a function with $\psi (n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$ E(ψ):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{ψ(n)}\sum_{j=1}^n\log a_j(x)=1\Big} $$ is completely determined without any extra condition on $\psi$.

10 pages

Cited by in corpus (3)

• Multifractal analysis of the convergence exponent in continued fractions
• Some exceptional sets of Borel-Bernstein Theorem in continued fractions
• On upper and lower fast Knintchine spectra of continued fractions