{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BIV3T3IL3YM2SAFOPKM7KSDS64","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":"d1536571a1de4b622eddf34a12eae033ec274cd52cd4865049f612c3634e715c","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-03-08T17:46:49Z","title_canon_sha256":"6652cbc5b9078bcf9ea5bc28844ff87e4417602a77fc652c28f20a2030a24355"},"schema_version":"1.0","source":{"id":"1703.02935","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.02935","created_at":"2026-05-18T00:02:02Z"},{"alias_kind":"arxiv_version","alias_value":"1703.02935v3","created_at":"2026-05-18T00:02:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.02935","created_at":"2026-05-18T00:02:02Z"},{"alias_kind":"pith_short_12","alias_value":"BIV3T3IL3YM2","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BIV3T3IL3YM2SAFO","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BIV3T3IL","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:7849569f5adc53b70ea9a27bf79d29e319273fc332ca11f9829314a338fda430","target":"graph","created_at":"2026-05-18T00:02:02Z","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":"Let $\\mu,\\nu$ be Radon measures on $\\mathbb{R}$, with $\\mu$ non-atomic and $\\nu$ doubling, and write $\\mu = \\mu_{a} + \\mu_{s}$ for the Lebesgue decomposition of $\\mu$ relative to $\\nu$. For an interval $I \\subset \\mathbb{R}$, define $\\alpha_{\\mu,\\nu}(I) := \\mathbb{W}_{1}(\\mu_{I},\\nu_{I})$, the Wasserstein distance of normalised blow-ups of $\\mu$ and $\\nu$ restricted to $I$. Let $\\mathcal{S}_{\\nu}$ be the square function $$\\mathcal{S}^{2}_{\\nu}(\\mu) = \\sum_{I \\in \\mathcal{D}} \\alpha_{\\mu,\\nu}^{2}(I)\\chi_{I},$$ where $\\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I p","authors_text":"Tuomas Orponen","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-03-08T17:46:49Z","title":"Absolute continuity and $\\alpha$-numbers on the real line"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02935","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:f986c3a86b5366bfd434370379bb85c62144c59c7271195d99caffd3597b7f32","target":"record","created_at":"2026-05-18T00:02:02Z","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":"d1536571a1de4b622eddf34a12eae033ec274cd52cd4865049f612c3634e715c","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-03-08T17:46:49Z","title_canon_sha256":"6652cbc5b9078bcf9ea5bc28844ff87e4417602a77fc652c28f20a2030a24355"},"schema_version":"1.0","source":{"id":"1703.02935","kind":"arxiv","version":3}},"canonical_sha256":"0a2bb9ed0bde19a900ae7a99f54872f72620f0fde8105e9dd2143a2a3b2bb6a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a2bb9ed0bde19a900ae7a99f54872f72620f0fde8105e9dd2143a2a3b2bb6a9","first_computed_at":"2026-05-18T00:02:02.272132Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:02.272132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s1s17jW6DBSyhFW8whN8ymAqI5kIUqYLSNodBpotGa/oIM73n8Yr4Witd0oo5EJkeVVy6B6OgEnpt2LTpxfNAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:02.272706Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.02935","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f986c3a86b5366bfd434370379bb85c62144c59c7271195d99caffd3597b7f32","sha256:7849569f5adc53b70ea9a27bf79d29e319273fc332ca11f9829314a338fda430"],"state_sha256":"764db566e994fcafe3fa6a53097ea18ec4e51c5ad534e56566668b81d09a37e9"}