Metrical properties of weighted products of consecutive L\"uroth digits

Abstract

The L\"uroth 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 $d_j\in\mathbb{N}_{\geq 2}$ for all $j\in\mathbb{N}$. Given $m\in \mathbb{N}$, $\mathbf{t}=(t_0,\ldots, t_{m-1})\in\mathbb{R}_{>0}^{m-1}$ and any function $\Psi:\mathbb{N}\to (1,\infty)$, define \[ \mathcal{E}_{\mathbf{t}}(\Psi)\colon= \left\{ x\in (0,1]: d_n^{t_0} \cdots d_{n+m}^{t_{m-1}}\geq \Psi(n) \text{ for infinitely many} \ n \in\mathbb{N} \right\}. \] We establish a Lebesgue measure dichotomy statement (a zero-one law) for $\mathcal{E}_{\mathbf{t}}(\Psi)$ under a natural non-removable condition $\liminf_{n\to\infty} \Psi(n)>~1$. Let $B$ be given by \[ \log B \colon= \liminf_{n\to\infty} \frac{\log(\Psi(n))}{n}. \] For any $m\in\mathbb{N}$, we compute the Hausdorff dimension of $\mathcal{E}_{\mathbf{t}}(\Psi)$ when either $B=1$ or $B=\infty$. We also compute the Hausdorff dimension of $\mathcal{E}_{\mathbf{t}}(\Psi)$ when $1<B< \infty$ for $m=2$.