  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.
