Multifractal analysis of maximal product of consecutive partial quotients in continued fractions

Let $[a_1(x),a_2(x),\dots ,a_n(x),\dots ]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. We study the growth rate of the maximal product of consecutive partial quotients among the first $n$ terms, defined by $L_n(x)={\max }_{1\le i\le n}\{a_i(x)a_{i+1}(x)\}$, from the viewpoint of multifractal analysis. More precisely, we determine the Hausdorff dimension of the level set $$L(\phi ):=\{x\in (0,1):\underset{n\to \mathrm{\infty }}{\lim }\frac{L_n(x)}{\phi (n)}=1\},$$ where $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is an increasing function such that $\log \phi$ is a regularly increasing function with index $\rho$. We show that there exists a jump of the Hausdorff dimension of $L(\phi )$ when $\rho =1\mathrm{/}2$. We also construct uncountably many discontinuous functions $\psi$ that cause the Hausdorff dimension of $L(\psi )$ to transition continuously from 1 to 1/2, filling the gap when $\rho =1\mathrm{/}2$.