Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue

TL;DR

This work analyzes the asymptotics of the $h_n$-dimension Hausdorff measure of the limit sets $J_n$ for the Gauss map $G$ and its piecewise-linear analogue. It proves an exact asymptotic formula for the linear model: $\displaystyle \lim_{n\to\infty} \frac{1-H_n}{1-h_n}\cdot \frac{1}{\ln n}=1$ and, equivalently, $\displaystyle \lim_{n\to\infty}\frac{n\,(1-H_n)}{\ln n}=\frac{1}{\chi}$ where $\chi=\sum_{m=1}^{\infty}\frac{\log(m(m+1))}{m(m+1)}$. For the nonlinear Gauss map, it establishes a sharp lower bound: $\displaystyle \liminf_{n\to\infty}\frac{1-H_n}{(1-h_n)\ln n}\ge 1$, which implies $\displaystyle \liminf_{n\to\infty}\frac{n\,(1-H_n)}{\ln n}\ge \frac{6}{\pi^2}$. The approach blends finite-IFS (via $J_n$) entropy methods, density theorems for Hausdorff measures, and distortion control (Koebe) to connect geometric measures with dynamical entropy and Lyapunov exponents. Overall, the results reveal how metric entropy governs the precise asymptotics of Cantor-type limit sets arising from continued-fraction dynamics, and they distinguish the linearized model from the nonlinear Gauss map in a concrete quantitative way.

Abstract

For $n\in\mathbb N$ we consider the set $J_n$ of points in the interval [0,1] whose continued fraction expansion entries are bounded by n. Similarly, we consider the set $J_n$ for the linearized analogue of the Gauss map. We study the asymptotic of the Hausdorff measure of the set $J_n$, (evaluated at its Hausdorff dimension). We obtain precise asymptotics for the linearized Gauss map and the same one -sided bound for the asymptotics for the original Gauss map.

Figures (2)

• Figure 1: First and second generation intervals.
• Figure 2: $I_1'$ compared to $I_1$

Theorems & Definitions (65)

• Definition 3.1
• Definition 3.3: and notation
• Theorem 4.2
• Corollary 4.3
• Corollary 4.4
• Proposition 4.5
• Proposition 4.6
• Lemma 5.1
• proof
• Proposition 5.2