Metrical properties of weighted products of consecutive Lüroth digits

The Lüroth expansion of a real number $x\in (0,1]$ is the series $$x=\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\frac{1}{d_1(d_1-1)d_2(d_2-1)d_3}+\cdots ,$$ with $d_j\in {\mathbb{N}}_{\ge 2}$ for all $j\in \mathbb{N}$. Given $m\in \mathbb{N}$, $\mathbf{t}=(t_0,\dots ,t_{m-1})\in {\mathbb{R}}_{>0}^{m-1}$ and any function $\mathrm{\Psi }:\mathbb{N}\to (1,\mathrm{\infty })$, define $${\mathcal{E}}_{\mathbf{t}}(\mathrm{\Psi })\text{\hspace{0.17em}⁣}:=\{x\in (0,1]:d_n^{t_0}\cdots d_{n+m}^{t_{m-1}}\ge \mathrm{\Psi }(n)\text{ for infinitely many }n\in \mathbb{N}\}\mathrm{.}$$ We establish a Lebesgue measure dichotomy statement (a zero-one law) for ${\mathcal{E}}_{\mathbf{t}}(\mathrm{\Psi })$ under a natural non-removable condition ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\mathrm{\Psi }(n)>1$. Let $B$ be given by $$\log B\text{\hspace{0.17em}⁣}:=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\log (\mathrm{\Psi }(n))}{n}\mathrm{.}$$ For any $m\in \mathbb{N}$, we compute the Hausdorff dimension of ${\mathcal{E}}_{\mathbf{t}}(\mathrm{\Psi })$ when either $B=1$ or $B=\mathrm{\infty }$. We also compute the Hausdorff dimension of ${\mathcal{E}}_{\mathbf{t}}(\mathrm{\Psi })$ when $1<B<\mathrm{\infty }$ for $m=2$.