GENERALIZATION OF SELBERG’S 3/16 THEOREM FOR CONVEX COCOMPACT THIN SUBGROUPS OF SO(n, 1)

The paper’s main theorem (uniform exponential mixing of the frame flow on congruence covers) is stated precisely in Theorem 1.3 and proved by introducing congruence transfer operators with holonomy, establishing uniform spectral bounds via a small-frequency expander argument plus a high-frequency Dolgopyat estimate, and then converting these bounds to correlation decay through a Paley–Wiener type inversion. This structure is explicit in the introduction and Sections 5 and 11, with the operator family M_{ξ,q,ρ} defined and analyzed in detail , and the overall plan summarized in the outline and concluding remarks on the Laplace transform method . The candidate solution outlines the same mechanism: fix a Markov section, define holonomy-and-congruence twisted Ruelle operators, combine an expander-based contraction at small |b| with a Dolgopyat-type estimate for large |b| or nontrivial holonomy, and invert the Laplace transform to obtain uniform exponential mixing with a polynomial factor in N_K(q); this aligns with the paper’s proofs, terminology, and hypotheses (including square-free levels coprime to q0 and strong approximation) and differs only in notation and minor presentational choices .