The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure

The paper proves explicit formulae for the Assouad and lower spectra of Kleinian limit sets and the associated Patterson–Sullivan (PS) measure (Theorems 3.1 and 3.2), using Stratmann–Velani’s global measure formula (GMF) and careful two-scale estimates for the escape function ρ, plus Bowditch-type geometry for certain set-spectrum bounds. The candidate solution derives exactly the same formulae, case by case, for dim_A^θ μ, dim_L^θ μ, dim_A^θ L(Γ), dim_L^θ L(Γ), and dim_B μ, and bases the argument on the GMF and two-scale window analysis. The expressions match the paper’s statements word-for-word, including the intermediate δ-regimes where both k_min and k_max appear. The paper’s proofs add technical lemmas (e.g., inequalities for ρ(z,Tθ)−ρ(z,T) and a Bowditch lemma for the set lower spectrum) that the candidate outlines only heuristically, but the overall approach is substantially the same and leads to the same results. See the paper’s statements and proofs of Theorems 3.1–3.2 and the GMF discussion for confirmation of all formulas and the methodology (GMF stated as Theorem 2.2; proofs in Section 4) .