{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:OQQXLNW2LQCE7HOYLT6ZBZGZYE","short_pith_number":"pith:OQQXLNW2","schema_version":"1.0","canonical_sha256":"742175b6da5c044f9dd85cfd90e4d9c13b782636e70ba104b7e97da56302ea37","source":{"kind":"arxiv","id":"1409.0222","version":1},"attestation_state":"computed","paper":{"title":"Rothberger gaps in fragmented ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Diego A. Mej\\'ia, J\\\"org Brendle","submitted_at":"2014-08-31T14:18:46Z","abstract_excerpt":"The~\\emph{Rothberger number} $\\mathfrak{b} (\\mathcal{I})$ of a definable ideal $\\mathcal{I}$ on $\\omega$ is the least cardinal $\\kappa$ such that there exists a Rothberger gap of type $(\\omega,\\kappa)$ in the quotient algebra $\\mathcal{P} (\\omega) / \\mathcal{I}$. We investigate $\\mathfrak{b} (\\mathcal{I})$ for a subclass of the $F_\\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consis"},"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":"1409.0222","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2014-08-31T14:18:46Z","cross_cats_sorted":[],"title_canon_sha256":"3858f369b601511cd5b234a6903b4b19b0dcfd19a42642c16416d7dbdf1e4593","abstract_canon_sha256":"062901536a50949140a958a1103e5b4fde7ae0cc806a61da9d316e0ff0ca37cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:53.143147Z","signature_b64":"9Ol/5d77qm5KPh1prYgNkoq2Smf5eiuH6/+7bzToe4w8WZF4YaGeLvDQyQcM8lsqWtWyIYOkBuHUTaMsWQRLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"742175b6da5c044f9dd85cfd90e4d9c13b782636e70ba104b7e97da56302ea37","last_reissued_at":"2026-05-18T02:43:53.142757Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:53.142757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rothberger gaps in fragmented ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Diego A. Mej\\'ia, J\\\"org Brendle","submitted_at":"2014-08-31T14:18:46Z","abstract_excerpt":"The~\\emph{Rothberger number} $\\mathfrak{b} (\\mathcal{I})$ of a definable ideal $\\mathcal{I}$ on $\\omega$ is the least cardinal $\\kappa$ such that there exists a Rothberger gap of type $(\\omega,\\kappa)$ in the quotient algebra $\\mathcal{P} (\\omega) / \\mathcal{I}$. We investigate $\\mathfrak{b} (\\mathcal{I})$ for a subclass of the $F_\\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0222","kind":"arxiv","version":1},"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":"1409.0222","created_at":"2026-05-18T02:43:53.142818+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.0222v1","created_at":"2026-05-18T02:43:53.142818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0222","created_at":"2026-05-18T02:43:53.142818+00:00"},{"alias_kind":"pith_short_12","alias_value":"OQQXLNW2LQCE","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"OQQXLNW2LQCE7HOY","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"OQQXLNW2","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/OQQXLNW2LQCE7HOYLT6ZBZGZYE","json":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE.json","graph_json":"https://pith.science/api/pith-number/OQQXLNW2LQCE7HOYLT6ZBZGZYE/graph.json","events_json":"https://pith.science/api/pith-number/OQQXLNW2LQCE7HOYLT6ZBZGZYE/events.json","paper":"https://pith.science/paper/OQQXLNW2"},"agent_actions":{"view_html":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE","download_json":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE.json","view_paper":"https://pith.science/paper/OQQXLNW2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.0222&json=true","fetch_graph":"https://pith.science/api/pith-number/OQQXLNW2LQCE7HOYLT6ZBZGZYE/graph.json","fetch_events":"https://pith.science/api/pith-number/OQQXLNW2LQCE7HOYLT6ZBZGZYE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE/action/storage_attestation","attest_author":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE/action/author_attestation","sign_citation":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE/action/citation_signature","submit_replication":"https://pith.science/pith/OQQXLNW2LQCE7HOYLT6ZBZGZYE/action/replication_record"}},"created_at":"2026-05-18T02:43:53.142818+00:00","updated_at":"2026-05-18T02:43:53.142818+00:00"}