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

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 $\{x\in(0,1]:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}2^{d_j(x)}=\xi\}$.