{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:D7CH45BQPDLRZP7BAWIO6GPSMA","short_pith_number":"pith:D7CH45BQ","schema_version":"1.0","canonical_sha256":"1fc47e743078d71cbfe10590ef19f2603aa4d0dda4829dc8529ee2b6ecf45520","source":{"kind":"arxiv","id":"1509.06294","version":2},"attestation_state":"computed","paper":{"title":"Rectifiability of harmonic measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alexander Volberg, Jonas Azzam, Jos\\'e Mar\\'ia Martell, Mihalis Mourgoglou, Steve Hofmann, Svitlana Mayboroda, Xavier Tolsa","submitted_at":"2015-09-21T16:38:05Z","abstract_excerpt":"In the present paper we prove that for any open connected set $\\Omega\\subset\\mathbb{R}^{n+1}$, $n\\geq 1$, and any $E\\subset \\partial \\Omega$ with $\\mathcal{H}^n(E)<\\infty$, absolute continuity of the harmonic measure $\\omega$ with respect to the Hausdorff measure on $E$ implies that $\\omega|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$."},"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":"1509.06294","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-21T16:38:05Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"52f118d5c392492b8dee837293783d3c771b64c62ecb3124907e6dbef8bf45f5","abstract_canon_sha256":"7a4d1aae8bf7996c9e0c8118e4e3992a561b043be6d23dae5249c9403abb0ce9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:50.620946Z","signature_b64":"SSK4Jt5Sk0ihVIg7PnaZtRmzNqGbzYta6IHRIlFmPCJfLs9K3o/lSBZsD7J89rT1ZBMC40+B2x9z0Ko003jsCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1fc47e743078d71cbfe10590ef19f2603aa4d0dda4829dc8529ee2b6ecf45520","last_reissued_at":"2026-05-18T00:03:50.620347Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:50.620347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rectifiability of harmonic measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alexander Volberg, Jonas Azzam, Jos\\'e Mar\\'ia Martell, Mihalis Mourgoglou, Steve Hofmann, Svitlana Mayboroda, Xavier Tolsa","submitted_at":"2015-09-21T16:38:05Z","abstract_excerpt":"In the present paper we prove that for any open connected set $\\Omega\\subset\\mathbb{R}^{n+1}$, $n\\geq 1$, and any $E\\subset \\partial \\Omega$ with $\\mathcal{H}^n(E)<\\infty$, absolute continuity of the harmonic measure $\\omega$ with respect to the Hausdorff measure on $E$ implies that $\\omega|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06294","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.06294","created_at":"2026-05-18T00:03:50.620449+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.06294v2","created_at":"2026-05-18T00:03:50.620449+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06294","created_at":"2026-05-18T00:03:50.620449+00:00"},{"alias_kind":"pith_short_12","alias_value":"D7CH45BQPDLR","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"D7CH45BQPDLRZP7B","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"D7CH45BQ","created_at":"2026-05-18T12:29:17.054201+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/D7CH45BQPDLRZP7BAWIO6GPSMA","json":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA.json","graph_json":"https://pith.science/api/pith-number/D7CH45BQPDLRZP7BAWIO6GPSMA/graph.json","events_json":"https://pith.science/api/pith-number/D7CH45BQPDLRZP7BAWIO6GPSMA/events.json","paper":"https://pith.science/paper/D7CH45BQ"},"agent_actions":{"view_html":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA","download_json":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA.json","view_paper":"https://pith.science/paper/D7CH45BQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.06294&json=true","fetch_graph":"https://pith.science/api/pith-number/D7CH45BQPDLRZP7BAWIO6GPSMA/graph.json","fetch_events":"https://pith.science/api/pith-number/D7CH45BQPDLRZP7BAWIO6GPSMA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA/action/storage_attestation","attest_author":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA/action/author_attestation","sign_citation":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA/action/citation_signature","submit_replication":"https://pith.science/pith/D7CH45BQPDLRZP7BAWIO6GPSMA/action/replication_record"}},"created_at":"2026-05-18T00:03:50.620449+00:00","updated_at":"2026-05-18T00:03:50.620449+00:00"}