Multifractal analysis of the power-2-decaying Gauss-like expansion - Wang, Xue-Jiao (Journal of mathematical analysis and applications, 2025)
Each real number x∈[0,1] admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as x=∑i∈N2−(d1(x)+d2(x)+⋯+di(x)), where di(x)∈N. For any x∈(0,1], the Khintchine exponent γ(x) is defined by γ(x):=limn→∞⁡1n∑j=1ndj(x) if the limit exists. We investigate the sizes of the level sets E(ξ):={x∈(0,1]:γ(x)=ξ} for ξ≥1. Utilizing the Ruelle operator theory, we obtain the Khintchine spectrum ξ↦dimH⁡E(ξ), where dimH 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]:limn→∞⁡1n∑j=1nlog⁡(dj(x))=ξ} and {x∈(0,1]:limn→∞⁡1n∑j=1n2dj(x)=ξ}.