Associated to the L\"{u}roth expansion of an irrational number $\bar{x}\in (0, 1)$ is a sequence $x=\{x_i\}\in {\mathbb N}^{\mathbb N}$. Then for each $n$, one can get an infinite probability vector $\Pi(x|n)=(p_{i,n})_{i\in \mathbb N}$ where $p_{i,n}$ is the frequency of $i$ occurring in the prefix of $\{x_i\}$ of length $n$. Let $ A(\{\Pi(x|n)\}_{n\in \mathbb N})$ be the set of accumulation points of the sequence $\{\Pi(x|n)\}_{n\in \mathbb N}$. Given a set $C$, let $$\Omega_{=C}=\left \{x \in {\mathbb N}^{\mathbb N}: A(\{\Pi(x|n)\}_{n\in \mathbb N})= C\right \}\;\;\text{and}\;\; \Omega_{\subseteq C}=\left\{x\in {\mathbb N}^{\mathbb N}: A(\{\Pi(x|n)\}_{n\in \mathbb N})\subseteq C\right\}.$$ In this talk, we mainly present the Hausdorff dimensions of $\Omega_{=C}$ and $\Omega_{\subseteq C}$. This is a joint work with Y. X. Gui and Y. Zhou.
