Exceptional sets related to the product of consecutive digits in Lüroth expansions

Every real number $x\in(0, 1]$ admits a Lüroth expansion $[d_{1}(x), d_{2}(x), \ldots]_{L}$ with $d_{n}(x)\in\mathbb{N}_{\ge 2}$ being its digits. Let $\big\{\frac{p_{n}(x)}{q_{n}(x)}, n\geq1 \big\}$ be the sequence of convergents of the Lüroth expansion of $x$. We study the growth rate of the product of consecutive digits relative to the denominator of the convergent for the Lüroth expansion of an irrational number. More precisely, given a natural number $m$, we prove that the set $$E_{m}(\beta)=\bigg\{x\in(0, 1]\colon \limsup_{n\rightarrow\infty}\frac{\log\big(d_{n}(x)d_{n+1}(x)\cdots d_{n+m}(x)\big)}{\log q_{n}(x)}=\beta\bigg\}$$ and the set $$\widetilde{E}_{m}(\beta)=\bigg\{x\in(0, 1]\colon \limsup_{n\rightarrow\infty}\frac{\log\big(d_{n}(x)d_{n+1}(x)\cdots d_{n+m}(x)\big)}{\log q_{n}(x)}\geq\beta\bigg\}$$ share the same Hausdorff dimension for $\beta\ge0$. It significantly generalises the existing results on the Hausdorff dimension of $E_{1}(\beta)$ and $\widetilde{E}_{1}(\beta)$.