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

Each real number x∈[0,1] admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as x=∑ᵢ 2-(d₁(x)+d₂(x)+⋯+dᵢ(x)), where dᵢ(x). For any x∈(0,1], the Khintchine exponent γ(x) is defined by γ(x):=limn→∞1/n∑ⱼ₌₁ⁿdⱼ(x) if the limit exists. We investigate the sizes of the level sets E(ξ):=∈(0,1]:γ(x)=ξ\ for ξ≥ 1. Utilizing the Ruelle operator theory, we obtain the Khintchine spectrum ξ↦H E(ξ), 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 ∈(0,1]:limn→∞1/n∑ⱼ₌₁ⁿlog(dⱼ(x))=ξ\ and ∈(0,1]:limn→∞1/n∑ⱼ₌₁ⁿ2dⱼ(x)=ξ\.