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Assume that $\\Omega^+$ is a $\\delta$-Reifenberg flat domain for some $\\delta>0$ small enough. In this paper we show that $\\log\\frac{d\\omega^-}{d\\omega^+}\\in VMO(\\omega^+)$ if and only if $\\Omega^+$ is vanishing Reifenberg flat, $\\Omega^+$ and $\\Omega^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $\\Omega^+$ has vanishing oscillation with respect to "},"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":"1904.00751","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-01T12:29:39Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"779023770334970a0cbefcd6608d12d1cb601f9e8e74382fc0288b04bc9e4178","abstract_canon_sha256":"d8503efbb0fddfb68d229fa0760e3fa8882ddb8180b6ea5baea446c65ab17af3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:49:07.413881Z","signature_b64":"rimZkq5G0tqYl3Cm6rf0bg71W0lkZCTJB2+nlTZX/qgJfH+rFdFrK85k5dUJF5FqNj1wNOv1NTPEyiyxhdaLDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27b7c2a0a6f772a28d499a73e16089b8e1e2635bfa4eebf290ba2529a3abc5f4","last_reissued_at":"2026-07-05T00:49:07.413495Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:49:07.413495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The two-phase problem for harmonic measure in VMO","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mart\\'i Prats, Xavier Tolsa","submitted_at":"2019-04-01T12:29:39Z","abstract_excerpt":"Let $\\Omega^+\\subset\\mathbb R^{n+1}$ be an NTA domain and let $\\Omega^-= \\mathbb R^{n+1}\\setminus \\overline{\\Omega^+}$ be an NTA domain as well. 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