{"paper":{"title":"Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\\operatorname{SO}(n, 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.SP"],"primary_cat":"math.DS","authors_text":"Pratyush Sarkar","submitted_at":"2026-06-17T04:20:23Z","abstract_excerpt":"Let $\\Gamma$ be a geometrically finite thin subgroup of an arithmetic lattice $\\Gamma_0 < G := \\operatorname{SO}(n, 1)$ and consider the congruence covers of $\\Gamma \\backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, \\Gamma_0) = (\\operatorname{SL}_2(\\mathbb{R}), \\operatorname{SL}_2(\\mathbb{Z}))$ when the critical exponent satisfies $\\delta_\\Gamma > \\frac{1}{2}$. The main applications are affine sieve for $\\Gamma$-orbits and uniform resonance-free half-planes for the resolvent of the Laplac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18674","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.18674/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}