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

TL;DR

This paper investigates the multifractal structure of the maximal product of consecutive partial quotients in continued fractions, determining the Hausdorff dimension of related level sets and revealing phase transitions at a critical growth rate.

Contribution

It provides a detailed multifractal analysis of the growth rate of maximal products of consecutive partial quotients, including the Hausdorff dimension of associated level sets and phase transition phenomena.

Findings

Identified the Hausdorff dimension of level sets based on growth rate functions.

Discovered a phase transition in Hausdorff dimension at a specific growth rate index.

Constructed functions causing continuous dimension transition from 1 to 1/2.

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 $\rho=1/2$.