{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:IFC6KQJNQANX6Z6VW7RMBTZ76I","short_pith_number":"pith:IFC6KQJN","canonical_record":{"source":{"id":"1304.4695","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-17T05:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"5423c996a8124bf3bdee8853b7d906212dcd3e758ebc61a8611bb93a8eeea8e6","abstract_canon_sha256":"5163ec5322ea8efb4cb0710578528c1b05dfe5ea81d71b709c05db4f2ba727b1"},"schema_version":"1.0"},"canonical_sha256":"4145e5412d801b7f67d5b7e2c0cf3ff23711d9926ebf7ce1fa0b51d3250e76d2","source":{"kind":"arxiv","id":"1304.4695","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4695","created_at":"2026-05-18T02:53:56Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4695v2","created_at":"2026-05-18T02:53:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4695","created_at":"2026-05-18T02:53:56Z"},{"alias_kind":"pith_short_12","alias_value":"IFC6KQJNQANX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IFC6KQJNQANX6Z6V","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IFC6KQJN","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:IFC6KQJNQANX6Z6VW7RMBTZ76I","target":"record","payload":{"canonical_record":{"source":{"id":"1304.4695","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-17T05:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"5423c996a8124bf3bdee8853b7d906212dcd3e758ebc61a8611bb93a8eeea8e6","abstract_canon_sha256":"5163ec5322ea8efb4cb0710578528c1b05dfe5ea81d71b709c05db4f2ba727b1"},"schema_version":"1.0"},"canonical_sha256":"4145e5412d801b7f67d5b7e2c0cf3ff23711d9926ebf7ce1fa0b51d3250e76d2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:56.426967Z","signature_b64":"bgat1W5WkNmSsKdjyQQPOgrRTwiWVYO0z+MX8xxCCx4GQdPQdf3RyP21ZUaQYiEbFbJHy68quOXT+gLj5u2QBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4145e5412d801b7f67d5b7e2c0cf3ff23711d9926ebf7ce1fa0b51d3250e76d2","last_reissued_at":"2026-05-18T02:53:56.426330Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:56.426330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.4695","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:53:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HxXrqmawEG9FbTH6bzKR9tAkW+QjJ61YeIQ/9tvxtJJbXNkwXeIO+JV9DiMtWPC5RGapOYpL0rKkwWU2Xem1AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:32:51.596083Z"},"content_sha256":"c5f8d8994dc18e7f3ab5d0df9a8004c327670024ea5b5616d5e3a75ef71cb4d7","schema_version":"1.0","event_id":"sha256:c5f8d8994dc18e7f3ab5d0df9a8004c327670024ea5b5616d5e3a75ef71cb4d7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:IFC6KQJNQANX6Z6VW7RMBTZ76I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Thickness conditions and Littlewood--Paley sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Vladimir Lebedev","submitted_at":"2013-04-17T05:34:41Z","abstract_excerpt":"We consider sets in the real line that have Littlewood--Paley properties $\\mathrm{LP}(p)$ or $\\mathrm{LP}$ and study the following question: How thick can these sets be?"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4695","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:53:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"anQr88pbnHOmNAQYYmB65iSkk68PhLsLyDTqCIulpyO2PFBhYRLr0X8H3yUcF0/T4Q9BfMvIr6h0FQCPInIECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:32:51.596437Z"},"content_sha256":"b4babeb659d4ae41cdebb74448b180d4b953078b5736f746d39b268ad3711e28","schema_version":"1.0","event_id":"sha256:b4babeb659d4ae41cdebb74448b180d4b953078b5736f746d39b268ad3711e28"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I/bundle.json","state_url":"https://pith.science/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I/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":"2026-06-01T13:32:51Z","links":{"resolver":"https://pith.science/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I","bundle":"https://pith.science/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I/bundle.json","state":"https://pith.science/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IFC6KQJNQANX6Z6VW7RMBTZ76I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:IFC6KQJNQANX6Z6VW7RMBTZ76I","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":"5163ec5322ea8efb4cb0710578528c1b05dfe5ea81d71b709c05db4f2ba727b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-17T05:34:41Z","title_canon_sha256":"5423c996a8124bf3bdee8853b7d906212dcd3e758ebc61a8611bb93a8eeea8e6"},"schema_version":"1.0","source":{"id":"1304.4695","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4695","created_at":"2026-05-18T02:53:56Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4695v2","created_at":"2026-05-18T02:53:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4695","created_at":"2026-05-18T02:53:56Z"},{"alias_kind":"pith_short_12","alias_value":"IFC6KQJNQANX","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IFC6KQJNQANX6Z6V","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IFC6KQJN","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:b4babeb659d4ae41cdebb74448b180d4b953078b5736f746d39b268ad3711e28","target":"graph","created_at":"2026-05-18T02:53:56Z","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 consider sets in the real line that have Littlewood--Paley properties $\\mathrm{LP}(p)$ or $\\mathrm{LP}$ and study the following question: How thick can these sets be?","authors_text":"Vladimir Lebedev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-17T05:34:41Z","title":"Thickness conditions and Littlewood--Paley sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4695","kind":"arxiv","version":2},"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:c5f8d8994dc18e7f3ab5d0df9a8004c327670024ea5b5616d5e3a75ef71cb4d7","target":"record","created_at":"2026-05-18T02:53:56Z","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":"5163ec5322ea8efb4cb0710578528c1b05dfe5ea81d71b709c05db4f2ba727b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-17T05:34:41Z","title_canon_sha256":"5423c996a8124bf3bdee8853b7d906212dcd3e758ebc61a8611bb93a8eeea8e6"},"schema_version":"1.0","source":{"id":"1304.4695","kind":"arxiv","version":2}},"canonical_sha256":"4145e5412d801b7f67d5b7e2c0cf3ff23711d9926ebf7ce1fa0b51d3250e76d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4145e5412d801b7f67d5b7e2c0cf3ff23711d9926ebf7ce1fa0b51d3250e76d2","first_computed_at":"2026-05-18T02:53:56.426330Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:56.426330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bgat1W5WkNmSsKdjyQQPOgrRTwiWVYO0z+MX8xxCCx4GQdPQdf3RyP21ZUaQYiEbFbJHy68quOXT+gLj5u2QBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:56.426967Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.4695","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5f8d8994dc18e7f3ab5d0df9a8004c327670024ea5b5616d5e3a75ef71cb4d7","sha256:b4babeb659d4ae41cdebb74448b180d4b953078b5736f746d39b268ad3711e28"],"state_sha256":"c6120a2f9afd4138a1d06900ccd7f28011f8511cb13e1657e8b8f1ea7c723854"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oGTcYgWk6CI3jhI0AiK2NT9KwgQGAZXz/MFBVnu7ugqtVWqCkiJorf3e6hE0A9PKzdrI+hfk4a847IG15hoPDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T13:32:51.598373Z","bundle_sha256":"b9f0e03f7081cb678064b71654fd13e17acaea5681a98ed9868bcc92f0e839bf"}}