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What proportion of the concentric circle $C_{\\epsilon} = \\partial B(z_0, r_0 + \\epsilon)$ is red, and what is the behavior of this quantity as $\\epsilon \\rightarrow 0$? Using equidistribution of closed horocycles on the modular surface $\\mathbb{H}^2/SL(2, \\mathbb{Z})$, we show that the answer is $\\frac{3}{\\pi} = 0.9549\\dots$ We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the"},"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":"1409.6352","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-22T21:57:42Z","cross_cats_sorted":[],"title_canon_sha256":"ed850f46ff623ee95aa19d59c83803708c8491bc15044304a3a6cbf8b75e5efe","abstract_canon_sha256":"5c623549a37db37d7644d8719d54f7f6789c1e8cbc409e1194a56fd05c0016cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:33:00.931740Z","signature_b64":"isS2USdE+LDBaXpfUKP4Qq9cib/poeihJdQi7Yp0dQeR6xF6pd4gXfX9+wDx0Ktur55nmIk6iKx+8LiN6sLYCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f60a02fed23ffcb118ae9f180a631142d5ff875c6fc885e65360b0cf6cfad22","last_reissued_at":"2026-05-18T02:33:00.931407Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:33:00.931407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Radial Density in Apollonian Packings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandru Zaharescu, Cristian Cobeli, Jayadev S. 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Using equidistribution of closed horocycles on the modular surface $\\mathbb{H}^2/SL(2, \\mathbb{Z})$, we show that the answer is $\\frac{3}{\\pi} = 0.9549\\dots$ We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6352","kind":"arxiv","version":2},"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":"1409.6352","created_at":"2026-05-18T02:33:00.931456+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.6352v2","created_at":"2026-05-18T02:33:00.931456+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.6352","created_at":"2026-05-18T02:33:00.931456+00:00"},{"alias_kind":"pith_short_12","alias_value":"P5QKAL7NEP74","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"P5QKAL7NEP74WEMK","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"P5QKAL7N","created_at":"2026-05-18T12:28:43.426989+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/P5QKAL7NEP74WEMK5HYYBJRRCQ","json":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ.json","graph_json":"https://pith.science/api/pith-number/P5QKAL7NEP74WEMK5HYYBJRRCQ/graph.json","events_json":"https://pith.science/api/pith-number/P5QKAL7NEP74WEMK5HYYBJRRCQ/events.json","paper":"https://pith.science/paper/P5QKAL7N"},"agent_actions":{"view_html":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ","download_json":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ.json","view_paper":"https://pith.science/paper/P5QKAL7N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.6352&json=true","fetch_graph":"https://pith.science/api/pith-number/P5QKAL7NEP74WEMK5HYYBJRRCQ/graph.json","fetch_events":"https://pith.science/api/pith-number/P5QKAL7NEP74WEMK5HYYBJRRCQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ/action/storage_attestation","attest_author":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ/action/author_attestation","sign_citation":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ/action/citation_signature","submit_replication":"https://pith.science/pith/P5QKAL7NEP74WEMK5HYYBJRRCQ/action/replication_record"}},"created_at":"2026-05-18T02:33:00.931456+00:00","updated_at":"2026-05-18T02:33:00.931456+00:00"}