On the maximal run-length function in the L\"uroth expansion

Let $\ell_n(x)$ denote the maximal run-length among the first $n$ digits of the L\"{u}roth expansion of $x\in(0,1]$. While $\ell_n(x)$ grows logarithmically, we investigate the finer multifractal properties of the exceptional set where $\ell_n(x)$ exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set $$\left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = \alpha, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = \beta \right\},$$ for all $0 \le \alpha \le \beta \le 1$.