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

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 $\gamma (x):={\lim }_{n\to \mathrm{\infty }}\frac{1}{n}{\sum }_{j=1}^n{d}_j(x)$ if the limit exists. We investigate the sizes of the level sets $E(\xi ):=\{x\in (0,1]:\gamma (x)=\xi \}$ for $\xi \ge 1$. 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 $\{x\in (0,1]:{\lim }_{n\to \mathrm{\infty }}\frac{1}{n}{\sum }_{j=1}^n\log (d_j(x))=\xi \}$ and $\{x\in (0,1]:{\lim }_{n\to \mathrm{\infty }}\frac{1}{n}{\sum }_{j=1}^n{2}^{d_j(x)}=\xi \}$.