{"paper":{"title":"Uniform congruence counting for Schottky semigroups in $\\mathrm{SL}_2(\\mathbf{Z})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Dale Winter, Hee Oh, Michael Magee","submitted_at":"2016-01-14T19:31:38Z","abstract_excerpt":"Let $\\Gamma$ be a Schottky semigroup in $\\mathrm{SL}_2(\\mathbf{Z})$, and for $q\\in \\mathbf N$, let $\\Gamma(q):=\\{\\gamma\\in \\Gamma: \\gamma= e \\text{ (mod $q$)}\\}$ be its congruence subsemigroup of level $q$. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls $B_R$ in $M_2(\\mathbf{R})$ of radius $R$: for all $q$ with no small prime factors, $ (\\Gamma (q) \\cap B_R )= c_\\Gamma \\frac{R^{2\\delta}}{ (\\mathrm{SL}_2(\\mathbf{Z}/q\\mathbf{Z}))} +O(q^C R^{2\\delta -\\epsilon})$ as $R\\to \\infty$ for some $c_\\Gamma >0, C>0, \\epsilon>0$ which are indepe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03705","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}