Multifractal analysis of maximal product of consecutive partial quotients in continued fractions (2502.02064v2)

Abstract: Let $[a_1(x), a_2(x), \ldots, a_n(x), \ldots]$ 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 $L_n(x)=\max_{1\leq i\leq n}{a_i(x)a_{i+1}(x)}$, from the viewpoint of multifractal analysis. More precisely, we determine the Hausdorff dimension of the level set [L(\varphi):=\left{x\in (0,1):\lim_{n\to \infty}\frac{L_n(x)}{\varphi(n)}=1\right},] where $\varphi:\mathbb{R^+}\to\mathbb{R^+}$ is an increasing function such that $\log \varphi$ is a regularly increasing function with index $\rho$. We show that there exists a jump of the Hausdorff dimension of $L(\varphi)$ when $\rho=1/2$. 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 $\rho=1/2$.