Full dimensional sets of reals whose sums of partial quotients increase in certain speed

Abstract

For a real $x\in (0,1)\setminus \mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots ]$ be its continued fraction expansion. Let $s_n(x)={\sum }_{j=1}^n{a}_j(x)$. The Hausdorff dimensions of the level sets $E_{\phi (n),\alpha }:=\{x\in (0,1):{\lim }_{n\to \infty }\frac{s_n(x)}{\phi (n)}=\alpha \}$ for $\alpha \ge 0$ and a non-decreasing sequence $\{\phi (n){\}}_{n=1}^{\infty }$ have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams \emph{et al}. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on $\phi (n)$ under which one can expect a $1$-dimensional set $E_{\phi (n),\alpha }$. We give certain upper and lower bounds on the increasing speed of $\phi (n)$ when $E_{\phi (n),\alpha }$ is of Hausdorff dimension 1 and a new class of sequences $\{\phi (n){\}}_{n=1}^{\infty }$ such that $E_{\phi (n),\alpha }$ is of full dimension. There is also a discussion of the problem in the irregular case.

References in corpus (3)

• Subexponentially increasing sums of partial quotients in continued fraction expansions
• Upper and lower fast Khintchine spectra in continued fractions
• A remark on the extreme value theory for continued fractions