{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:QR7FPN43PYPAJ242QIKLERKXTW","short_pith_number":"pith:QR7FPN43","canonical_record":{"source":{"id":"1304.3943","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-14T20:03:19Z","cross_cats_sorted":[],"title_canon_sha256":"7d2cd95ecd7baf2aad61dadbc5fc0e0bca8ad19ea58ab90100e21f983a5e03a0","abstract_canon_sha256":"48084f408211a895e4caa67bcc59fe012316049a1b9645ee7ddc7221ba11e00c"},"schema_version":"1.0"},"canonical_sha256":"847e57b79b7e1e04eb9a8214b245579db268db82fecd74ca52808887e3f92e49","source":{"kind":"arxiv","id":"1304.3943","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.3943","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1304.3943v2","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.3943","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"QR7FPN43PYPA","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QR7FPN43PYPAJ242","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QR7FPN43","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:QR7FPN43PYPAJ242QIKLERKXTW","target":"record","payload":{"canonical_record":{"source":{"id":"1304.3943","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-14T20:03:19Z","cross_cats_sorted":[],"title_canon_sha256":"7d2cd95ecd7baf2aad61dadbc5fc0e0bca8ad19ea58ab90100e21f983a5e03a0","abstract_canon_sha256":"48084f408211a895e4caa67bcc59fe012316049a1b9645ee7ddc7221ba11e00c"},"schema_version":"1.0"},"canonical_sha256":"847e57b79b7e1e04eb9a8214b245579db268db82fecd74ca52808887e3f92e49","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:35.612685Z","signature_b64":"jAfxnMtrZC9gGGz/TmLCcoM+jsqJgimwhwmNrPtRe+of2FxObdfh4VVrypp9/2ip7/Epa5inMWQSWywskJixBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"847e57b79b7e1e04eb9a8214b245579db268db82fecd74ca52808887e3f92e49","last_reissued_at":"2026-05-18T03:05:35.612085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:35.612085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.3943","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":"2026-05-18T03:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1uKv0MdKL1C3TST2mGiQKJODPSF2m+zvClpvyJrLKW5klJWCkKVsYoMiVX3YIMXvObQj0iYDGWWcWC8YgsW+CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:39:46.972708Z"},"content_sha256":"287982c084b4de72811e1df442aecafc18a70c87e4d9271a426a311b966353a2","schema_version":"1.0","event_id":"sha256:287982c084b4de72811e1df442aecafc18a70c87e4d9271a426a311b966353a2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:QR7FPN43PYPAJ242QIKLERKXTW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lacunary Fourier and Walsh-Fourier series near L^1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Francesco Di Plinio","submitted_at":"2013-04-14T20:03:19Z","abstract_excerpt":"We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.3943","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":"2026-05-18T03:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GnUZux2GnfpBkdYlIeooWptxQlLZ3//K07WlJA5GOIVlLUdINF6Me/fwD+TooZjQFay6h2C+FNgjZahPInNsAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:39:46.973240Z"},"content_sha256":"908b5c491a87bbbf5aa56d311e38a5571bbc061ced467ea7a235a65d0d46c739","schema_version":"1.0","event_id":"sha256:908b5c491a87bbbf5aa56d311e38a5571bbc061ced467ea7a235a65d0d46c739"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QR7FPN43PYPAJ242QIKLERKXTW/bundle.json","state_url":"https://pith.science/pith/QR7FPN43PYPAJ242QIKLERKXTW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QR7FPN43PYPAJ242QIKLERKXTW/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":"2026-06-05T09:39:46Z","links":{"resolver":"https://pith.science/pith/QR7FPN43PYPAJ242QIKLERKXTW","bundle":"https://pith.science/pith/QR7FPN43PYPAJ242QIKLERKXTW/bundle.json","state":"https://pith.science/pith/QR7FPN43PYPAJ242QIKLERKXTW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QR7FPN43PYPAJ242QIKLERKXTW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:QR7FPN43PYPAJ242QIKLERKXTW","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":"48084f408211a895e4caa67bcc59fe012316049a1b9645ee7ddc7221ba11e00c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-14T20:03:19Z","title_canon_sha256":"7d2cd95ecd7baf2aad61dadbc5fc0e0bca8ad19ea58ab90100e21f983a5e03a0"},"schema_version":"1.0","source":{"id":"1304.3943","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.3943","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1304.3943v2","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.3943","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"QR7FPN43PYPA","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QR7FPN43PYPAJ242","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QR7FPN43","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:908b5c491a87bbbf5aa56d311e38a5571bbc061ced467ea7a235a65d0d46c739","target":"graph","created_at":"2026-05-18T03:05:35Z","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 prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for","authors_text":"Francesco Di Plinio","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-14T20:03:19Z","title":"Lacunary Fourier and Walsh-Fourier series near L^1"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.3943","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":"sha256:287982c084b4de72811e1df442aecafc18a70c87e4d9271a426a311b966353a2","target":"record","created_at":"2026-05-18T03:05:35Z","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":"48084f408211a895e4caa67bcc59fe012316049a1b9645ee7ddc7221ba11e00c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-14T20:03:19Z","title_canon_sha256":"7d2cd95ecd7baf2aad61dadbc5fc0e0bca8ad19ea58ab90100e21f983a5e03a0"},"schema_version":"1.0","source":{"id":"1304.3943","kind":"arxiv","version":2}},"canonical_sha256":"847e57b79b7e1e04eb9a8214b245579db268db82fecd74ca52808887e3f92e49","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"847e57b79b7e1e04eb9a8214b245579db268db82fecd74ca52808887e3f92e49","first_computed_at":"2026-05-18T03:05:35.612085Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:35.612085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jAfxnMtrZC9gGGz/TmLCcoM+jsqJgimwhwmNrPtRe+of2FxObdfh4VVrypp9/2ip7/Epa5inMWQSWywskJixBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:35.612685Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.3943","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:287982c084b4de72811e1df442aecafc18a70c87e4d9271a426a311b966353a2","sha256:908b5c491a87bbbf5aa56d311e38a5571bbc061ced467ea7a235a65d0d46c739"],"state_sha256":"b7323dd684b786769b2e5d0cdae26a41b4300d3370a6560b41c3133d5886afe4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"419qb/OULn94oMJn+KUPP+MPhu89nUn+0z2jwue3kdIYuZFIdN0F1KEeCHKkeZEBHdJQT9Ibsh5S5vaR58MLAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T09:39:46.975497Z","bundle_sha256":"cbbdeb7240a30ef4220b2fed96f631a960954ede8b2dacfab95dfbceb2697ee0"}}