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

Let ${\mathrm{\ell }}_n(x)$ denote the maximal run-length among the first $n$ digits of the L"{u}roth expansion of $x\in (0,1]$. While ${\mathrm{\ell }}_n(x)$ grows logarithmically, we investigate the finer multifractal properties of the exceptional set where ${\mathrm{\ell }}_n(x)$ exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set $$\{x\in (0,1]:\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{{\mathrm{\ell }}_n(x)}{n}=\alpha ,\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{{\mathrm{\ell }}_n(x)}{n}=\beta \},$$ for all $0\le \alpha \le \beta \le 1$.