{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:6LAQQBLCGZCZTFEBS4VJLNH5O3","short_pith_number":"pith:6LAQQBLC","canonical_record":{"source":{"id":"1410.2782","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-10-10T13:43:08Z","cross_cats_sorted":["math.AP","math.MG"],"title_canon_sha256":"aff77d948b5057742764a0b5eb4c3a313883b68df2508591bb13a75b7b9227ce","abstract_canon_sha256":"fd08c18b6c283829f19bdd4ca63313b6a2a48d1f68c321a2396ec55af1806ef0"},"schema_version":"1.0"},"canonical_sha256":"f2c10805623645999481972a95b4fd76e1b4a3b225a928ed77fb0e69ec66411a","source":{"kind":"arxiv","id":"1410.2782","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.2782","created_at":"2026-05-18T01:19:30Z"},{"alias_kind":"arxiv_version","alias_value":"1410.2782v3","created_at":"2026-05-18T01:19:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.2782","created_at":"2026-05-18T01:19:30Z"},{"alias_kind":"pith_short_12","alias_value":"6LAQQBLCGZCZ","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6LAQQBLCGZCZTFEB","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6LAQQBLC","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:6LAQQBLCGZCZTFEBS4VJLNH5O3","target":"record","payload":{"canonical_record":{"source":{"id":"1410.2782","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-10-10T13:43:08Z","cross_cats_sorted":["math.AP","math.MG"],"title_canon_sha256":"aff77d948b5057742764a0b5eb4c3a313883b68df2508591bb13a75b7b9227ce","abstract_canon_sha256":"fd08c18b6c283829f19bdd4ca63313b6a2a48d1f68c321a2396ec55af1806ef0"},"schema_version":"1.0"},"canonical_sha256":"f2c10805623645999481972a95b4fd76e1b4a3b225a928ed77fb0e69ec66411a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:30.651513Z","signature_b64":"WBE0uOqjKNZ0Dm9G68Gmp+MGlUCiHkODGvr4jReXMrCcpizz8ws0NPU7a6NmSz423Emx1uOLirCKAP5cz4ZEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2c10805623645999481972a95b4fd76e1b4a3b225a928ed77fb0e69ec66411a","last_reissued_at":"2026-05-18T01:19:30.650924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:30.650924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.2782","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:19:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g9nyQng5PHtdGxT6a6FzQHc/EIgkHKgIPupZlpvfSYGssINR9z5NTIbtteuT/J4X/3gVMuwau014bpDYR89kCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:11:14.949994Z"},"content_sha256":"95ba7c5742bbc84d813f5958e764ed80b82384407d549647d7198560cc8c6f18","schema_version":"1.0","event_id":"sha256:95ba7c5742bbc84d813f5958e764ed80b82384407d549647d7198560cc8c6f18"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:6LAQQBLCGZCZTFEBS4VJLNH5O3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sets of absolute continuity for harmonic measure in NTA domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.CA","authors_text":"Jonas Azzam","submitted_at":"2014-10-10T13:43:08Z","abstract_excerpt":"We show that if $\\Omega$ is an NTA domain with harmonic measure $w$ and $E\\subseteq \\partial\\Omega$ is contained in an Ahlfors regular set, then $w|_{E}\\ll \\mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that for all $\\tau>0$ $w$ obeys an $A_{\\infty}$-type condition with respect to $\\mathscr{H}^{d}|_{E'}$, where $E'\\subseteq E$ is so that $w(E\\backslash E')<\\tau w(E)$, even though $\\partial\\Omega$ may not even be locally $\\mathscr{H}^{d}$-finite. We also show that, for uniform domains with uniform complements, if $E\\subseteq\\partial\\Omega$ is the Lipschitz image of a su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.2782","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:19:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GVdF/KBtsu34zYdf1KGj+U2G6XNGz3HWxJWrZ1FqWjCxkrqnoDhn+yMjf8uUOkF64HX8VNsmCikgWOMZW41UAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:11:14.950734Z"},"content_sha256":"14471e87de603b55cde87df97717ecc87d0ce0b500e55ee5056119de8e4f867b","schema_version":"1.0","event_id":"sha256:14471e87de603b55cde87df97717ecc87d0ce0b500e55ee5056119de8e4f867b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3/bundle.json","state_url":"https://pith.science/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T18:11:14Z","links":{"resolver":"https://pith.science/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3","bundle":"https://pith.science/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3/bundle.json","state":"https://pith.science/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6LAQQBLCGZCZTFEBS4VJLNH5O3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6LAQQBLCGZCZTFEBS4VJLNH5O3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fd08c18b6c283829f19bdd4ca63313b6a2a48d1f68c321a2396ec55af1806ef0","cross_cats_sorted":["math.AP","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-10-10T13:43:08Z","title_canon_sha256":"aff77d948b5057742764a0b5eb4c3a313883b68df2508591bb13a75b7b9227ce"},"schema_version":"1.0","source":{"id":"1410.2782","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.2782","created_at":"2026-05-18T01:19:30Z"},{"alias_kind":"arxiv_version","alias_value":"1410.2782v3","created_at":"2026-05-18T01:19:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.2782","created_at":"2026-05-18T01:19:30Z"},{"alias_kind":"pith_short_12","alias_value":"6LAQQBLCGZCZ","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6LAQQBLCGZCZTFEB","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6LAQQBLC","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:14471e87de603b55cde87df97717ecc87d0ce0b500e55ee5056119de8e4f867b","target":"graph","created_at":"2026-05-18T01:19:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that if $\\Omega$ is an NTA domain with harmonic measure $w$ and $E\\subseteq \\partial\\Omega$ is contained in an Ahlfors regular set, then $w|_{E}\\ll \\mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that for all $\\tau>0$ $w$ obeys an $A_{\\infty}$-type condition with respect to $\\mathscr{H}^{d}|_{E'}$, where $E'\\subseteq E$ is so that $w(E\\backslash E')<\\tau w(E)$, even though $\\partial\\Omega$ may not even be locally $\\mathscr{H}^{d}$-finite. We also show that, for uniform domains with uniform complements, if $E\\subseteq\\partial\\Omega$ is the Lipschitz image of a su","authors_text":"Jonas Azzam","cross_cats":["math.AP","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-10-10T13:43:08Z","title":"Sets of absolute continuity for harmonic measure in NTA domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.2782","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:95ba7c5742bbc84d813f5958e764ed80b82384407d549647d7198560cc8c6f18","target":"record","created_at":"2026-05-18T01:19:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fd08c18b6c283829f19bdd4ca63313b6a2a48d1f68c321a2396ec55af1806ef0","cross_cats_sorted":["math.AP","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-10-10T13:43:08Z","title_canon_sha256":"aff77d948b5057742764a0b5eb4c3a313883b68df2508591bb13a75b7b9227ce"},"schema_version":"1.0","source":{"id":"1410.2782","kind":"arxiv","version":3}},"canonical_sha256":"f2c10805623645999481972a95b4fd76e1b4a3b225a928ed77fb0e69ec66411a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2c10805623645999481972a95b4fd76e1b4a3b225a928ed77fb0e69ec66411a","first_computed_at":"2026-05-18T01:19:30.650924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:30.650924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WBE0uOqjKNZ0Dm9G68Gmp+MGlUCiHkODGvr4jReXMrCcpizz8ws0NPU7a6NmSz423Emx1uOLirCKAP5cz4ZEBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:30.651513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.2782","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95ba7c5742bbc84d813f5958e764ed80b82384407d549647d7198560cc8c6f18","sha256:14471e87de603b55cde87df97717ecc87d0ce0b500e55ee5056119de8e4f867b"],"state_sha256":"a81a0a8ff99603f80ba97edc6dc95fafdee78e29b8e3e2f08cdcb538008f940c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cbWUXKC24+q7Ntfv0QrGInNGxn9XRjFLAtO5vRo91a+STTQKrHtbweZiQ34lk1TsHajIA2KMtGge2sky4sUsCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T18:11:14.954643Z","bundle_sha256":"f14c8b0b838a9fe6ff23576d005d399b441a70fb25c03f0a41891d3d673e7d47"}}