{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:N2SFLK25EC6TQSJQWXFDDGRG35","short_pith_number":"pith:N2SFLK25","canonical_record":{"source":{"id":"1403.7502","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-28T19:23:41Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"d7b57c50e1cf0b7d542dc70cc94c0585aa059f662e4de45e505b1937f9de745b","abstract_canon_sha256":"7548ed227579276fa9149c265393256b4ebab0fbc57df13d8800f5305658f136"},"schema_version":"1.0"},"canonical_sha256":"6ea455ab5d20bd384930b5ca319a26df7826b65aa8f56023e07a8caee2621ccf","source":{"kind":"arxiv","id":"1403.7502","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.7502","created_at":"2026-05-18T01:28:14Z"},{"alias_kind":"arxiv_version","alias_value":"1403.7502v2","created_at":"2026-05-18T01:28:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7502","created_at":"2026-05-18T01:28:14Z"},{"alias_kind":"pith_short_12","alias_value":"N2SFLK25EC6T","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"N2SFLK25EC6TQSJQ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"N2SFLK25","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:N2SFLK25EC6TQSJQWXFDDGRG35","target":"record","payload":{"canonical_record":{"source":{"id":"1403.7502","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-28T19:23:41Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"d7b57c50e1cf0b7d542dc70cc94c0585aa059f662e4de45e505b1937f9de745b","abstract_canon_sha256":"7548ed227579276fa9149c265393256b4ebab0fbc57df13d8800f5305658f136"},"schema_version":"1.0"},"canonical_sha256":"6ea455ab5d20bd384930b5ca319a26df7826b65aa8f56023e07a8caee2621ccf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:14.996974Z","signature_b64":"gsy1ilO+BalhStdP0JKYsImlRvgMu4pgs17zxBdEWzzwfbDFde5AMzLqb+rKQASBXxdKRV0PuJj7EbDNY1p+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ea455ab5d20bd384930b5ca319a26df7826b65aa8f56023e07a8caee2621ccf","last_reissued_at":"2026-05-18T01:28:14.996285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:14.996285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1403.7502","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:28:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3ITPaoOZx4qRqAvQV9yI8vfuTtC6S+4W9dnRWPMD5QdllUhy+fMPAuD7PIwfbZFWtnvd6pH5APsOgLyCjRlHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:23:52.760626Z"},"content_sha256":"6c1d578de6928dbbd1465554174e3d848d31f9b2ac5ab4c849231e9cd98d77c8","schema_version":"1.0","event_id":"sha256:6c1d578de6928dbbd1465554174e3d848d31f9b2ac5ab4c849231e9cd98d77c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:N2SFLK25EC6TQSJQWXFDDGRG35","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Poincar\\'e sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Byron Heersink","submitted_at":"2014-03-28T19:23:41Z","abstract_excerpt":"For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincar\\'e section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of SL(2,R)/SL(2,Z). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erd\\H{o}s, Sz\\\"usz, and Tur\\'an."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7502","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:28:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hJG7bsvLCr2vlDEc3m4BV9PdmJe1fsZorqRW41EBJKx5i0EjSvJIrMKRjbLNfSGI9bmKe7dTgA5lb8J5tLeIAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:23:52.760957Z"},"content_sha256":"4517163cbbef72026b6fecc0c265ecb38a1d10a036ad4b1326d77c2d911a0854","schema_version":"1.0","event_id":"sha256:4517163cbbef72026b6fecc0c265ecb38a1d10a036ad4b1326d77c2d911a0854"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N2SFLK25EC6TQSJQWXFDDGRG35/bundle.json","state_url":"https://pith.science/pith/N2SFLK25EC6TQSJQWXFDDGRG35/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N2SFLK25EC6TQSJQWXFDDGRG35/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T22:23:52Z","links":{"resolver":"https://pith.science/pith/N2SFLK25EC6TQSJQWXFDDGRG35","bundle":"https://pith.science/pith/N2SFLK25EC6TQSJQWXFDDGRG35/bundle.json","state":"https://pith.science/pith/N2SFLK25EC6TQSJQWXFDDGRG35/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N2SFLK25EC6TQSJQWXFDDGRG35/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:N2SFLK25EC6TQSJQWXFDDGRG35","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7548ed227579276fa9149c265393256b4ebab0fbc57df13d8800f5305658f136","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-28T19:23:41Z","title_canon_sha256":"d7b57c50e1cf0b7d542dc70cc94c0585aa059f662e4de45e505b1937f9de745b"},"schema_version":"1.0","source":{"id":"1403.7502","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.7502","created_at":"2026-05-18T01:28:14Z"},{"alias_kind":"arxiv_version","alias_value":"1403.7502v2","created_at":"2026-05-18T01:28:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7502","created_at":"2026-05-18T01:28:14Z"},{"alias_kind":"pith_short_12","alias_value":"N2SFLK25EC6T","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"N2SFLK25EC6TQSJQ","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"N2SFLK25","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:4517163cbbef72026b6fecc0c265ecb38a1d10a036ad4b1326d77c2d911a0854","target":"graph","created_at":"2026-05-18T01:28:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincar\\'e section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of SL(2,R)/SL(2,Z). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erd\\H{o}s, Sz\\\"usz, and Tur\\'an.","authors_text":"Byron Heersink","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-28T19:23:41Z","title":"Poincar\\'e sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7502","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6c1d578de6928dbbd1465554174e3d848d31f9b2ac5ab4c849231e9cd98d77c8","target":"record","created_at":"2026-05-18T01:28:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7548ed227579276fa9149c265393256b4ebab0fbc57df13d8800f5305658f136","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-28T19:23:41Z","title_canon_sha256":"d7b57c50e1cf0b7d542dc70cc94c0585aa059f662e4de45e505b1937f9de745b"},"schema_version":"1.0","source":{"id":"1403.7502","kind":"arxiv","version":2}},"canonical_sha256":"6ea455ab5d20bd384930b5ca319a26df7826b65aa8f56023e07a8caee2621ccf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ea455ab5d20bd384930b5ca319a26df7826b65aa8f56023e07a8caee2621ccf","first_computed_at":"2026-05-18T01:28:14.996285Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:28:14.996285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gsy1ilO+BalhStdP0JKYsImlRvgMu4pgs17zxBdEWzzwfbDFde5AMzLqb+rKQASBXxdKRV0PuJj7EbDNY1p+CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:28:14.996974Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.7502","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c1d578de6928dbbd1465554174e3d848d31f9b2ac5ab4c849231e9cd98d77c8","sha256:4517163cbbef72026b6fecc0c265ecb38a1d10a036ad4b1326d77c2d911a0854"],"state_sha256":"c76bee1043fe8437ef7a3983fe34f034794cbb01eb4db8844cd78b306264691b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MakEoB7UoIsUEKk7DBaHH+IdlmktIPJL8J9yafcPCB9am9toHbFOSUPZ4YSqqscEcmTwDDSLGdKfsXNA/jDsCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:23:52.762742Z","bundle_sha256":"837b6d5b113703788d893a5c64924dcf5157df435dec341dac807b9fb5ba8d00"}}