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 $E_{φ(n),α}:=\{x\in(0,1): \lim_{n\rightarrow\infty}\frac{s_n(x)}{φ(n)}=α\}$ for $α\geq 0$ and a non-decreasing sequence $\{φ(n)\}_{n=1}^\infty$ 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 $φ(n)$ under which one can expect a $1$-dimensional set $E_{φ(n),α}$. We give certain upper and lower bounds on the increasing speed of $φ(n)$ when $E_{φ(n),α}$ is of Hausdorff dimension 1 and a new class of sequences $\{φ(n)\}_{n=1}^\infty$ such that $E_{φ(n),α}$ is of full dimension. There is also a discussion of the problem in the irregular case.