{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:3A4DSPZN6QSRR5QK6324MX57ZN","short_pith_number":"pith:3A4DSPZN","schema_version":"1.0","canonical_sha256":"d838393f2df42518f60af6f5c65fbfcb58de8465d1aa857bcaf8581c5ee32cf5","source":{"kind":"arxiv","id":"2010.09551","version":2},"attestation_state":"computed","paper":{"title":"A topological approach to undefinability in algebraic extensions of $\\mathbb{Q}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Caleb Springer, Kirsten Eisentraeger, Linda Westrick, Russell Miller","submitted_at":"2020-10-19T14:28:19Z","abstract_excerpt":"For any subset $Z \\subseteq \\mathbb{Q}$, consider the set $S_Z$ of subfields $L\\subseteq \\overline{\\mathbb{Q}}$ which contain a co-infinite subset $C \\subseteq L$ that is universally definable in $L$ such that $C \\cap \\mathbb{Q}=Z$. Placing a natural topology on the set $\\text{Sub}(\\overline{\\mathbb{Q}})$ of subfields of $\\overline{\\mathbb{Q}}$, we show that if $Z$ is not thin in $\\mathbb{Q}$, then $S_Z$ is meager in $\\text{Sub}(\\overline{\\mathbb{Q}})$. Here, thin and meager both mean \"small\", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a mea"},"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":"2010.09551","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2020-10-19T14:28:19Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"ba45b1d2ae104d1c7fbad126dfbc9360a12fac6aff3345b7a976f9e8d50d1a0f","abstract_canon_sha256":"041c3437e5e475ab1fe30e468123d6a4da1b84414bd171efb209356dc1d680d4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:05:37.923162Z","signature_b64":"sxMyXIoABZJDtBtzFN9PC46qUK0dqM2VN9to4uD5C4rXXTBUJNtAzI31vzhgK6os4FqaIGdANn4VFVMaGPXiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d838393f2df42518f60af6f5c65fbfcb58de8465d1aa857bcaf8581c5ee32cf5","last_reissued_at":"2026-07-05T07:05:37.922739Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:05:37.922739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A topological approach to undefinability in algebraic extensions of $\\mathbb{Q}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Caleb Springer, Kirsten Eisentraeger, Linda Westrick, Russell Miller","submitted_at":"2020-10-19T14:28:19Z","abstract_excerpt":"For any subset $Z \\subseteq \\mathbb{Q}$, consider the set $S_Z$ of subfields $L\\subseteq \\overline{\\mathbb{Q}}$ which contain a co-infinite subset $C \\subseteq L$ that is universally definable in $L$ such that $C \\cap \\mathbb{Q}=Z$. Placing a natural topology on the set $\\text{Sub}(\\overline{\\mathbb{Q}})$ of subfields of $\\overline{\\mathbb{Q}}$, we show that if $Z$ is not thin in $\\mathbb{Q}$, then $S_Z$ is meager in $\\text{Sub}(\\overline{\\mathbb{Q}})$. Here, thin and meager both mean \"small\", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a mea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.09551","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2010.09551/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":"2010.09551","created_at":"2026-07-05T07:05:37.922792+00:00"},{"alias_kind":"arxiv_version","alias_value":"2010.09551v2","created_at":"2026-07-05T07:05:37.922792+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.09551","created_at":"2026-07-05T07:05:37.922792+00:00"},{"alias_kind":"pith_short_12","alias_value":"3A4DSPZN6QSR","created_at":"2026-07-05T07:05:37.922792+00:00"},{"alias_kind":"pith_short_16","alias_value":"3A4DSPZN6QSRR5QK","created_at":"2026-07-05T07:05:37.922792+00:00"},{"alias_kind":"pith_short_8","alias_value":"3A4DSPZN","created_at":"2026-07-05T07:05:37.922792+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/3A4DSPZN6QSRR5QK6324MX57ZN","json":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN.json","graph_json":"https://pith.science/api/pith-number/3A4DSPZN6QSRR5QK6324MX57ZN/graph.json","events_json":"https://pith.science/api/pith-number/3A4DSPZN6QSRR5QK6324MX57ZN/events.json","paper":"https://pith.science/paper/3A4DSPZN"},"agent_actions":{"view_html":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN","download_json":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN.json","view_paper":"https://pith.science/paper/3A4DSPZN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2010.09551&json=true","fetch_graph":"https://pith.science/api/pith-number/3A4DSPZN6QSRR5QK6324MX57ZN/graph.json","fetch_events":"https://pith.science/api/pith-number/3A4DSPZN6QSRR5QK6324MX57ZN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN/action/storage_attestation","attest_author":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN/action/author_attestation","sign_citation":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN/action/citation_signature","submit_replication":"https://pith.science/pith/3A4DSPZN6QSRR5QK6324MX57ZN/action/replication_record"}},"created_at":"2026-07-05T07:05:37.922792+00:00","updated_at":"2026-07-05T07:05:37.922792+00:00"}