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Let $\Gamma$ be a convex cocompact thin subgroup of an arithmetic lattice in$\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem inthis setting, i.e., we prove uniform exponential mixing of the frame flow andobtain a uniform resonance-free half plane for the congruence covers of thehyperbolic manifold $\Gamma \backslash \mathbb H^n$. This extends the work ofOh-Winter who established the $n = 2$ case. The theorem follows from uniformspectral bounds for the congruence transfer operators with holonomy. We employSarkar-Winter's frame flow version of Dolgopyat's method uniformly over thecongruence covers as well as Golsefidy-Varj\'{u}'s generalization ofBourgain-Gamburd-Sarnak's expansion machinery by using the properties that thereturn trajectory subgroups are Zariski dense and have trace fields whichcoincide with that of $\Gamma$. These properties follow by proving that thereturn trajectory subgroups have finite index in $\Gamma$.

Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$