Some exceptional sets of Borel-Bernstein Theorem in continued fractions

Abstract

Let $[a_1(x),a_2(x),a_3(x),\cdots ]$ denote the continued fraction expansion of a real number $x\in [0,1)$. This paper is concerned with certain exceptional sets of the Borel-Bernstein Theorem on the growth rate of $\{a_n(x){\}}_{n\ge 1}$. As a main result, the Hausdorff dimension of the set \[ E_{\sup}(ψ)=\left\{x\in[0,1):\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\} \] is determined, where $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$ tends to infinity as $n\to \infty$.

15 pages

References in corpus (3)

• On the fast Khintchine spectrum in continued fractions
• A remark on the extreme value theory for continued fractions
• Multifractal analysis of the convergence exponent in continued fractions