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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.03705","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-14T19:31:38Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"de7eb87b36603ca1e631655b5cf8f266e010317d9166907a39215405531078ae","abstract_canon_sha256":"32c080b2ea534dc6d7bdcc77d28d1b02da43f4e635b9237ece959d9a49df12fd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:51.891759Z","signature_b64":"FitMW6aH3sxQu/uT/8PzHn7Khik/ISh/I050dTkIe+DCZxpRTshfuf3dDkm0Tj4asN906uTJpsW4fQzTtN/iAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5c9a4eab9616d2ccc5f1e9af49ab89afc7567cab9ec8b514844142dcebd157a","last_reissued_at":"2026-05-18T00:35:51.891254Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:51.891254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1601.03705","created_at":"2026-05-18T00:35:51.891330+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.03705v3","created_at":"2026-05-18T00:35:51.891330+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03705","created_at":"2026-05-18T00:35:51.891330+00:00"},{"alias_kind":"pith_short_12","alias_value":"6XE2J2VZMFWS","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"6XE2J2VZMFWSZTC7","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"6XE2J2VZ","created_at":"2026-05-18T12:30:04.600751+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL","json":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL.json","graph_json":"https://pith.science/api/pith-number/6XE2J2VZMFWSZTC7D2NPJGVYTL/graph.json","events_json":"https://pith.science/api/pith-number/6XE2J2VZMFWSZTC7D2NPJGVYTL/events.json","paper":"https://pith.science/paper/6XE2J2VZ"},"agent_actions":{"view_html":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL","download_json":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL.json","view_paper":"https://pith.science/paper/6XE2J2VZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.03705&json=true","fetch_graph":"https://pith.science/api/pith-number/6XE2J2VZMFWSZTC7D2NPJGVYTL/graph.json","fetch_events":"https://pith.science/api/pith-number/6XE2J2VZMFWSZTC7D2NPJGVYTL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL/action/storage_attestation","attest_author":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL/action/author_attestation","sign_citation":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL/action/citation_signature","submit_replication":"https://pith.science/pith/6XE2J2VZMFWSZTC7D2NPJGVYTL/action/replication_record"}},"created_at":"2026-05-18T00:35:51.891330+00:00","updated_at":"2026-05-18T00:35:51.891330+00:00"}