{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:R43EMQK4HXJF2WNP6HRHUQHPCP","short_pith_number":"pith:R43EMQK4","schema_version":"1.0","canonical_sha256":"8f3646415c3dd25d59aff1e27a40ef13df1e84768bcdc3e44df7523331089f5e","source":{"kind":"arxiv","id":"1512.05966","version":1},"attestation_state":"computed","paper":{"title":"Universal and complete sets in martingale theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.LO","authors_text":"Dominique Lecomte, Miroslav Zeleny","submitted_at":"2015-12-18T14:10:55Z","abstract_excerpt":"The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\\_{\\delta\\sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is ${\\bf\\Pi}^1\\_1$-complete, in a uniform way. We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes ${\\bf\\"},"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":"1512.05966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-12-18T14:10:55Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"9642ed6083726338bdb95637c0d1a3c22d9d56a2c13cf26166779ae145665b55","abstract_canon_sha256":"18f564ad2e5affc0bb62ad79d97042a8bb2c5e73fcbcc13b8870c507f26ac96d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:05.774496Z","signature_b64":"c8Bd+unogC+EiWftEMfVIajKNzJyC2denpyPENJUGzbtL7quUT4R7+868dq3cnExXzE5CwErC2Rmh5SXF9FRDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f3646415c3dd25d59aff1e27a40ef13df1e84768bcdc3e44df7523331089f5e","last_reissued_at":"2026-05-18T01:24:05.773779Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:05.773779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal and complete sets in martingale theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.LO","authors_text":"Dominique Lecomte, Miroslav Zeleny","submitted_at":"2015-12-18T14:10:55Z","abstract_excerpt":"The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\\_{\\delta\\sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is ${\\bf\\Pi}^1\\_1$-complete, in a uniform way. We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes ${\\bf\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05966","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":"1512.05966","created_at":"2026-05-18T01:24:05.773901+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.05966v1","created_at":"2026-05-18T01:24:05.773901+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05966","created_at":"2026-05-18T01:24:05.773901+00:00"},{"alias_kind":"pith_short_12","alias_value":"R43EMQK4HXJF","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"R43EMQK4HXJF2WNP","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"R43EMQK4","created_at":"2026-05-18T12:29:39.896362+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/R43EMQK4HXJF2WNP6HRHUQHPCP","json":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP.json","graph_json":"https://pith.science/api/pith-number/R43EMQK4HXJF2WNP6HRHUQHPCP/graph.json","events_json":"https://pith.science/api/pith-number/R43EMQK4HXJF2WNP6HRHUQHPCP/events.json","paper":"https://pith.science/paper/R43EMQK4"},"agent_actions":{"view_html":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP","download_json":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP.json","view_paper":"https://pith.science/paper/R43EMQK4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.05966&json=true","fetch_graph":"https://pith.science/api/pith-number/R43EMQK4HXJF2WNP6HRHUQHPCP/graph.json","fetch_events":"https://pith.science/api/pith-number/R43EMQK4HXJF2WNP6HRHUQHPCP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP/action/storage_attestation","attest_author":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP/action/author_attestation","sign_citation":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP/action/citation_signature","submit_replication":"https://pith.science/pith/R43EMQK4HXJF2WNP6HRHUQHPCP/action/replication_record"}},"created_at":"2026-05-18T01:24:05.773901+00:00","updated_at":"2026-05-18T01:24:05.773901+00:00"}