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

Each real number x0, 1 admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as x=₈ 2^- (d₁ (x) +d₂ (x) ++dᵢ (x) ), where dᵢ (x). For any x (0, 1], the Khintchine exponent (x) is defined by (x): =₍1n₉=₁ⁿdⱼ (x) if the limit exists. We investigate the sizes of the level sets E (): =\x (0, 1]: (x) =\ for 1. Utilizing the Ruelle operator theory, we obtain the Khintchine spectrum H E (), where 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 (0, 1]: ₍1{n₉=₁^n (dⱼ (x) ) =\} and \x (0, 1]: ₍1{n₉=₁^n2^dⱼ (x) =\}.