### The fractional dimensional theory in Lüroth expansion

It is well known that every $x\in (0,1]$ can be expanded to an infinite Lüroth series in the form of $$x=\frac{1}{{d}_{1}\left(x\right)}+\cdots +\frac{1}{{d}_{1}\left(x\right)({d}_{1}\left(x\right)-1)\cdots {d}_{n-1}\left(x\right)({d}_{n-1}\left(x\right)-1){d}_{n}\left(x\right)}+\cdots ,$$ where ${d}_{n}\left(x\right)\ge 2$ for all $n\ge 1$. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $${F}_{\phi}=\{x\in (0,1]:{d}_{n}\left(x\right)\ge \phi \left(n\right),\phantom{\rule{4pt}{0ex}}\forall n\ge 1\}$$ are completely determined, where $\phi $ is an integer-valued function defined on $\mathbb{N}$, and $\phi \left(n\right)\to \infty $ as $n\to \infty $.