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

The paper proves, via thermodynamic formalism (explicit pressure and Gibbs measures), that for the P2GLE map T(x)=2^n x−1 on (2^{-n},2^{-(n-1)}], the Khintchine spectrum is dim_H E(ξ) = [log(ξ−1)/ξ + log ξ − log(ξ−1)]/log 2 for ξ>1 and dim_H E(1)=0, with properties (1)–(5) including t(2)=1, limits to 0, monotonicity change at ξ=2, and a unique inflection point (Theorem 1.1 and Theorem 3.1) . The pressure is computed explicitly as P(t,q)=log(e^{-t log 2 + q}/(1−e^{-t log 2 + q})), yielding q(ξ)=log(ξ−1)/ξ and the stated t(ξ) by solving P(t,q)=qξ and ∂P/∂q=ξ (Eq. (3.3)–(3.4)) . The boundary case dim_H E(1)=0 is established by an appropriate covering argument (Proposition 3.4) . The candidate solution derives the same spectrum via a different, elementary route: coding by the full shift, a geometric Bernoulli push-forward for the lower bound, and sharp combinatorial counting (compositions) plus Stirling for the upper bound; this matches the paper’s formula and properties. Thus both are correct, with different proof strategies.