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 $E_{ \varphi(n),\alpha}:=\{ x\in(0,1): \lim_{ n\rightarrow\infty}\frac{ s_n(x)}{ \varphi(n)}=\alpha\}$ for $\alpha\geq 0$ and a non-decreasing sequence $\{ \varphi(n)\}_{ n=1}^\infty$ have been studied by E. Cesaratto, B. Vall\'ee, 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 $\varphi(n)$ under which one can expect a $1$-dimensional set $E_{ \varphi(n),\alpha}$. We give certain upper and lower bounds on the increasing speed of $\varphi(n)$ when $E_{ \varphi(n),\alpha}$ is of Hausdorff dimension 1 and a new class of sequences $\{ \varphi(n)\}_{ n=1}^\infty$ such that $E_{ \varphi(n),\alpha}$ is of full dimension. There is also a discussion of the problem in the irregular case.
Categories: math.NT