{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:BVWLDYIM4T5CU6MOESTDQAQEAX","short_pith_number":"pith:BVWLDYIM","canonical_record":{"source":{"id":"1402.7094","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-27T21:40:36Z","cross_cats_sorted":[],"title_canon_sha256":"609787e9d21a1e383efa2d45580fa1c3e13715528a1e2cd93652d682a2dd3196","abstract_canon_sha256":"c70667bea22f5f745fd8adbb01ac8270f0c991178f92af94c304c8c8c8666bd3"},"schema_version":"1.0"},"canonical_sha256":"0d6cb1e10ce4fa2a798e24a638020405d2224981681fbdc2ce21f12aec43b5d6","source":{"kind":"arxiv","id":"1402.7094","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.7094","created_at":"2026-05-18T02:52:57Z"},{"alias_kind":"arxiv_version","alias_value":"1402.7094v2","created_at":"2026-05-18T02:52:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.7094","created_at":"2026-05-18T02:52:57Z"},{"alias_kind":"pith_short_12","alias_value":"BVWLDYIM4T5C","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"BVWLDYIM4T5CU6MO","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"BVWLDYIM","created_at":"2026-05-18T12:28:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:BVWLDYIM4T5CU6MOESTDQAQEAX","target":"record","payload":{"canonical_record":{"source":{"id":"1402.7094","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-27T21:40:36Z","cross_cats_sorted":[],"title_canon_sha256":"609787e9d21a1e383efa2d45580fa1c3e13715528a1e2cd93652d682a2dd3196","abstract_canon_sha256":"c70667bea22f5f745fd8adbb01ac8270f0c991178f92af94c304c8c8c8666bd3"},"schema_version":"1.0"},"canonical_sha256":"0d6cb1e10ce4fa2a798e24a638020405d2224981681fbdc2ce21f12aec43b5d6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:57.686444Z","signature_b64":"8hA3wyQlH0u/s0V5YrOWCMicXSnn+5qTlgh+iLFqYqIVohYiaFS1foc2QBZ0Fc99HswIDgrHKe+ODEERfn9EBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d6cb1e10ce4fa2a798e24a638020405d2224981681fbdc2ce21f12aec43b5d6","last_reissued_at":"2026-05-18T02:52:57.685780Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:57.685780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.7094","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":"2026-05-18T02:52:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ke7ZTkGuEG2htmxqB5oi3nqvpOkxKtAVi7jJxAzL0HSgXv14V79JF6GAd1W7PxDre8lmbLX97h07cmjZBsWHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:17:54.684228Z"},"content_sha256":"8db2ca7d9574a290d53abd6cfd0d4197b2213b4c0ba7a8fd01565d7e7ef8a89c","schema_version":"1.0","event_id":"sha256:8db2ca7d9574a290d53abd6cfd0d4197b2213b4c0ba7a8fd01565d7e7ef8a89c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:BVWLDYIM4T5CU6MOESTDQAQEAX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pointwise characteristic factors for Wiener Wintner double recurrence theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"David Duncan, Idris Assani, Ryo Moore","submitted_at":"2014-02-27T21:40:36Z","abstract_excerpt":"In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let $(X, \\mathcal{F}, \\mu, T)$ be a standard ergodic system. We will show that for any $f_1, f_2 \\in L^\\infty(X)$, the double recurrence Wiener-Wintner average\n  \\[ \\frac{1}{N} \\sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x) e^{2\\pi i n t} \\] converges off a single null set of $X$ independent of $t$ as $N \\to \\infty$. Furthermore, we will show a uniform Wiener-Wintner double recurrence result: If either $f_1$ or $f_2$ belongs to the orthogonal complement of the Conze-Lesigne factor, then there exists a set of fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.7094","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":"2026-05-18T02:52:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VJPrx6HWxiUgRsMjuDHDvv9h9b/2loeGbSoTthth7rFXoRuo+/sAOmM6B+sJLxw5K0pxRW2SGeV76JjOIra9Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:17:54.684606Z"},"content_sha256":"1fc2d93af0addd3078ad090f4b133b764b6c13e4bb2729521b3696760aefd761","schema_version":"1.0","event_id":"sha256:1fc2d93af0addd3078ad090f4b133b764b6c13e4bb2729521b3696760aefd761"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BVWLDYIM4T5CU6MOESTDQAQEAX/bundle.json","state_url":"https://pith.science/pith/BVWLDYIM4T5CU6MOESTDQAQEAX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BVWLDYIM4T5CU6MOESTDQAQEAX/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":"2026-06-23T14:17:54Z","links":{"resolver":"https://pith.science/pith/BVWLDYIM4T5CU6MOESTDQAQEAX","bundle":"https://pith.science/pith/BVWLDYIM4T5CU6MOESTDQAQEAX/bundle.json","state":"https://pith.science/pith/BVWLDYIM4T5CU6MOESTDQAQEAX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BVWLDYIM4T5CU6MOESTDQAQEAX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:BVWLDYIM4T5CU6MOESTDQAQEAX","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":"c70667bea22f5f745fd8adbb01ac8270f0c991178f92af94c304c8c8c8666bd3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-27T21:40:36Z","title_canon_sha256":"609787e9d21a1e383efa2d45580fa1c3e13715528a1e2cd93652d682a2dd3196"},"schema_version":"1.0","source":{"id":"1402.7094","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.7094","created_at":"2026-05-18T02:52:57Z"},{"alias_kind":"arxiv_version","alias_value":"1402.7094v2","created_at":"2026-05-18T02:52:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.7094","created_at":"2026-05-18T02:52:57Z"},{"alias_kind":"pith_short_12","alias_value":"BVWLDYIM4T5C","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"BVWLDYIM4T5CU6MO","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"BVWLDYIM","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:1fc2d93af0addd3078ad090f4b133b764b6c13e4bb2729521b3696760aefd761","target":"graph","created_at":"2026-05-18T02:52:57Z","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":"In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let $(X, \\mathcal{F}, \\mu, T)$ be a standard ergodic system. We will show that for any $f_1, f_2 \\in L^\\infty(X)$, the double recurrence Wiener-Wintner average\n  \\[ \\frac{1}{N} \\sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x) e^{2\\pi i n t} \\] converges off a single null set of $X$ independent of $t$ as $N \\to \\infty$. Furthermore, we will show a uniform Wiener-Wintner double recurrence result: If either $f_1$ or $f_2$ belongs to the orthogonal complement of the Conze-Lesigne factor, then there exists a set of fu","authors_text":"David Duncan, Idris Assani, Ryo Moore","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-27T21:40:36Z","title":"Pointwise characteristic factors for Wiener Wintner double recurrence theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.7094","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":"sha256:8db2ca7d9574a290d53abd6cfd0d4197b2213b4c0ba7a8fd01565d7e7ef8a89c","target":"record","created_at":"2026-05-18T02:52:57Z","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":"c70667bea22f5f745fd8adbb01ac8270f0c991178f92af94c304c8c8c8666bd3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-02-27T21:40:36Z","title_canon_sha256":"609787e9d21a1e383efa2d45580fa1c3e13715528a1e2cd93652d682a2dd3196"},"schema_version":"1.0","source":{"id":"1402.7094","kind":"arxiv","version":2}},"canonical_sha256":"0d6cb1e10ce4fa2a798e24a638020405d2224981681fbdc2ce21f12aec43b5d6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d6cb1e10ce4fa2a798e24a638020405d2224981681fbdc2ce21f12aec43b5d6","first_computed_at":"2026-05-18T02:52:57.685780Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:52:57.685780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8hA3wyQlH0u/s0V5YrOWCMicXSnn+5qTlgh+iLFqYqIVohYiaFS1foc2QBZ0Fc99HswIDgrHKe+ODEERfn9EBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:52:57.686444Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.7094","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8db2ca7d9574a290d53abd6cfd0d4197b2213b4c0ba7a8fd01565d7e7ef8a89c","sha256:1fc2d93af0addd3078ad090f4b133b764b6c13e4bb2729521b3696760aefd761"],"state_sha256":"7777234746dacf4930c689b0e9f5cf727c954024ceca117e401852bdf8c7893c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QP3jeUYD3G9ECtu9JSE4SwXNdzvIVLCJg1pk5Zq/ac9sEypNrUw6Bp2ViPjPRep8YXZdNKO7kpjBKJxUcCXfDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T14:17:54.686829Z","bundle_sha256":"747ac00173cd86b37a031f1b0da4273fee0e8c5ef04856fd5e27ecaff7074e56"}}