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Boca, Jack Anderson","submitted_at":"2025-08-11T13:07:13Z","abstract_excerpt":"The sequence $({\\mathscr S}_Q)_Q$ of $\\operatorname{SL}(2,{\\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\\mathscr S}_Q := \\{ a/q \\in {\\mathbb Q} \\cap (0,1]: q+a+\\bar{a} \\le Q\\}$, where $\\bar{a}$ is the multiplicative inverse of $a\\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\\mathscr S}_Q$ is dense in the region ${\\mathcal V}:=\\{ (x,y)\\in [0,1]^2 : \\max \\{ (1-3x)/2,2x-1\\} \\le y \\le \\max \\{ x,1-x\\} \\}$, and provide a formula for their distribution in ${\\mathcal V}$ as $Q\\rightarrow \\infty$."},"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":"2508.07951","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-08-11T13:07:13Z","cross_cats_sorted":[],"title_canon_sha256":"b6ee9cc30a7ce4f29bb4bfe5a16e6c25a6f9976e951511c3afad42b9e02d3cec","abstract_canon_sha256":"7d2a638da315c4c11767d2fc9c42d4e7e772f4be3d3a30f189e43ea0a25e0880"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:02:53.016023Z","signature_b64":"dLy9EgBM8M8e7+ibyuScj3iZ+r/TPDzQUEkACP9EuO/X2qOSs0YsmMSwzftBBNueuNSlmR+rRSwEHSDghjdJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5928498f54c5f5d901bb40b7962f7f7e76fc254d43cb0ae680d1ca2c19ec8ce2","last_reissued_at":"2026-05-20T00:02:53.015092Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:02:53.015092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On denominators of consecutive $\\operatorname{SL}(2,{\\mathbb N})$-saturated Farey fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandru Zaharescu, Cristian Cobeli, Florin P. Boca, Jack Anderson","submitted_at":"2025-08-11T13:07:13Z","abstract_excerpt":"The sequence $({\\mathscr S}_Q)_Q$ of $\\operatorname{SL}(2,{\\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\\mathscr S}_Q := \\{ a/q \\in {\\mathbb Q} \\cap (0,1]: q+a+\\bar{a} \\le Q\\}$, where $\\bar{a}$ is the multiplicative inverse of $a\\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\\mathscr S}_Q$ is dense in the region ${\\mathcal V}:=\\{ (x,y)\\in [0,1]^2 : \\max \\{ (1-3x)/2,2x-1\\} \\le y \\le \\max \\{ x,1-x\\} \\}$, and provide a formula for their distribution in ${\\mathcal V}$ as $Q\\rightarrow \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07951","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.07951/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2508.07951","created_at":"2026-05-20T00:02:53.015246+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.07951v3","created_at":"2026-05-20T00:02:53.015246+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.07951","created_at":"2026-05-20T00:02:53.015246+00:00"},{"alias_kind":"pith_short_12","alias_value":"LEUETD2UYX25","created_at":"2026-05-20T00:02:53.015246+00:00"},{"alias_kind":"pith_short_16","alias_value":"LEUETD2UYX25SAN3","created_at":"2026-05-20T00:02:53.015246+00:00"},{"alias_kind":"pith_short_8","alias_value":"LEUETD2U","created_at":"2026-05-20T00:02:53.015246+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/LEUETD2UYX25SAN3IC3ZML37PZ","json":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ.json","graph_json":"https://pith.science/api/pith-number/LEUETD2UYX25SAN3IC3ZML37PZ/graph.json","events_json":"https://pith.science/api/pith-number/LEUETD2UYX25SAN3IC3ZML37PZ/events.json","paper":"https://pith.science/paper/LEUETD2U"},"agent_actions":{"view_html":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ","download_json":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ.json","view_paper":"https://pith.science/paper/LEUETD2U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.07951&json=true","fetch_graph":"https://pith.science/api/pith-number/LEUETD2UYX25SAN3IC3ZML37PZ/graph.json","fetch_events":"https://pith.science/api/pith-number/LEUETD2UYX25SAN3IC3ZML37PZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ/action/storage_attestation","attest_author":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ/action/author_attestation","sign_citation":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ/action/citation_signature","submit_replication":"https://pith.science/pith/LEUETD2UYX25SAN3IC3ZML37PZ/action/replication_record"}},"created_at":"2026-05-20T00:02:53.015246+00:00","updated_at":"2026-05-20T00:02:53.015246+00:00"}