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From this result we deduce that a compact set $E\\subset\\mathbb R^{n+1}$ with $H^n(E)<\\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions."},"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":"1212.5431","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-21T13:43:30Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"7c36bcb92b340efae8e046f7dbbf69babf6897f8fea71e5198c56a68c159e0cc","abstract_canon_sha256":"d58e5f09229ac2cca589c711ea7d1e7e6e28d093e152ff1ff0f68f64e39e6e78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:28.316541Z","signature_b64":"Gw1yDAyb5LmSM3Kc7vpOgjebX9OEG4M4fKMnFn4xuKJ4cdksoHXRfe6x+5Oj4MnYZ/EvhIsMCeOIod4SgGxTAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"445ee5e4d6d2288853686f08371d856049408c4cccea13f271f1a3dcd6088f70","last_reissued_at":"2026-05-18T03:05:28.315993Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:28.315993Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alexander Volberg, Fedor Nazarov, Xavier Tolsa","submitted_at":"2012-12-21T13:43:30Z","abstract_excerpt":"We show that, given a set $E\\subset \\mathbb R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \\int_{E} \\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\\subset\\mathbb R^{n+1}$ with $H^n(E)<\\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5431","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":"1212.5431","created_at":"2026-05-18T03:05:28.316079+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.5431v2","created_at":"2026-05-18T03:05:28.316079+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5431","created_at":"2026-05-18T03:05:28.316079+00:00"},{"alias_kind":"pith_short_12","alias_value":"IRPOLZGW2IUI","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_16","alias_value":"IRPOLZGW2IUIQU3I","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_8","alias_value":"IRPOLZGW","created_at":"2026-05-18T12:27:09.501522+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/IRPOLZGW2IUIQU3IN4EDOHMFMB","json":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB.json","graph_json":"https://pith.science/api/pith-number/IRPOLZGW2IUIQU3IN4EDOHMFMB/graph.json","events_json":"https://pith.science/api/pith-number/IRPOLZGW2IUIQU3IN4EDOHMFMB/events.json","paper":"https://pith.science/paper/IRPOLZGW"},"agent_actions":{"view_html":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB","download_json":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB.json","view_paper":"https://pith.science/paper/IRPOLZGW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.5431&json=true","fetch_graph":"https://pith.science/api/pith-number/IRPOLZGW2IUIQU3IN4EDOHMFMB/graph.json","fetch_events":"https://pith.science/api/pith-number/IRPOLZGW2IUIQU3IN4EDOHMFMB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB/action/storage_attestation","attest_author":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB/action/author_attestation","sign_citation":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB/action/citation_signature","submit_replication":"https://pith.science/pith/IRPOLZGW2IUIQU3IN4EDOHMFMB/action/replication_record"}},"created_at":"2026-05-18T03:05:28.316079+00:00","updated_at":"2026-05-18T03:05:28.316079+00:00"}