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For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ beits continued fraction expansion. Let $s_n(x)=\sum_{j=1}^n a_j(x)$. TheHausdorff 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 studiedby 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 theirwork, that is, we consider the conditions on $\varphi(n)$ under which one canexpect a $1$-dimensional set $E_{\varphi(n),\alpha}$. We give certain upper andlower bounds on the increasing speed of $\varphi(n)$ when$E_{\varphi(n),\alpha}$ is of Hausdorff dimension 1 and a new class ofsequences $\{\varphi(n)\}_{n=1}^\infty$ such that $E_{\varphi(n),\alpha}$ is offull dimension. There is also a discussion of the problem in the irregularcase.

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