Multifractal analysis of the power-2-decaying Gauss-like expansion

TL;DR

This paper studies the multifractal structure of a unique power-2-decaying expansion for real numbers, deriving the Hausdorff dimension spectrum of level sets related to the Khintchine exponent and revealing a unique inflection point.

Contribution

It introduces the Khintchine spectrum for the P2GLE, proves its unique inflection point, and connects it to the Lyapunov spectrum, advancing understanding of multifractal analysis in this context.

Findings

The Khintchine spectrum has exactly one inflection point.

Explicit Hausdorff dimensions are obtained for various level sets.

The spectrum's properties differ from those in continued fractions.

Abstract

Each real number $x\in[0,1]$ admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as $x=\sum_{i\in\mathbb{N}} 2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))}$, where $d_i(x)\in\mathbb{N}$. For any $x\in(0,1]$, the Khintchine exponent $\gamma(x)$ is defined by $\gamma(x):=\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^nd_j(x)$ if the limit exists. We investigate the sizes of the level sets $E(\xi):=\{x\in(0,1]:\gamma(x)=\xi\}$ for $\xi\geq 1$. Utilizing the Ruelle operator theory, we obtain the Khintchine spectrum $\xi\mapsto\dim_H E(\xi)$, where $\dim_H$ denotes the Hausdorff dimension. We establish the remarkable fact that the Khintchine spectrum has exactly one inflection point, which was never proved for the corresponding spectrum in continued fractions. As a direct consequence, we also obtain the Lyapunov spectrum. Furthermore, we find the Hausdorff dimensions of the level sets $\{x\in(0,1]:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\log(d_j(x))=\xi\}$ and $\{x\in(0,1]:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}2^{d_j(x)}=\xi\}$.