{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:5PKRM7QQBZ5LSBTM3EGNPYMXHT","short_pith_number":"pith:5PKRM7QQ","canonical_record":{"source":{"id":"1409.0463","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-25T02:09:18Z","cross_cats_sorted":[],"title_canon_sha256":"1fd0c000916fe7979df4afcb0eb37f0078b40b0787d4ef2cd37fc7903cd27e9c","abstract_canon_sha256":"5d2cd16a66ed4ff10ae778104e26a160c2c2f7a9268caafd5c0a3a2c9ad04fff"},"schema_version":"1.0"},"canonical_sha256":"ebd5167e100e7ab9066cd90cd7e1973cf94b984cf5314019be890cd2379d7e4e","source":{"kind":"arxiv","id":"1409.0463","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0463","created_at":"2026-05-18T01:34:05Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0463v2","created_at":"2026-05-18T01:34:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0463","created_at":"2026-05-18T01:34:05Z"},{"alias_kind":"pith_short_12","alias_value":"5PKRM7QQBZ5L","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5PKRM7QQBZ5LSBTM","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5PKRM7QQ","created_at":"2026-05-18T12:28:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:5PKRM7QQBZ5LSBTM3EGNPYMXHT","target":"record","payload":{"canonical_record":{"source":{"id":"1409.0463","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-25T02:09:18Z","cross_cats_sorted":[],"title_canon_sha256":"1fd0c000916fe7979df4afcb0eb37f0078b40b0787d4ef2cd37fc7903cd27e9c","abstract_canon_sha256":"5d2cd16a66ed4ff10ae778104e26a160c2c2f7a9268caafd5c0a3a2c9ad04fff"},"schema_version":"1.0"},"canonical_sha256":"ebd5167e100e7ab9066cd90cd7e1973cf94b984cf5314019be890cd2379d7e4e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:05.573691Z","signature_b64":"UJggvj7YYld8amTytjq/xw928zsw3L1uzp3M1VN0rqqFfsMvoUJP2RS33uxfAVY7SJgfIw/36nbOwyPIvojJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebd5167e100e7ab9066cd90cd7e1973cf94b984cf5314019be890cd2379d7e4e","last_reissued_at":"2026-05-18T01:34:05.573072Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:05.573072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.0463","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":"2026-05-18T01:34:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fPsX7Sm4GuiVdAwhbL45GvaGXqiKkkDyJa0X3uN5iwPGmQ7X+tXoLIdsADWq6vdXImKDD/9Ey8qJD+kY9rEiAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T10:04:06.598608Z"},"content_sha256":"0ffe1667ae7724d8687b83186c05f8bb7d071cbcec41556973250036cd00cf7a","schema_version":"1.0","event_id":"sha256:0ffe1667ae7724d8687b83186c05f8bb7d071cbcec41556973250036cd00cf7a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:5PKRM7QQBZ5LSBTM3EGNPYMXHT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extension of Wiener-Wintner double recurrence theorem to polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Idris Assani, Ryo Moore","submitted_at":"2014-08-25T02:09:18Z","abstract_excerpt":"We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages \\[ \\frac{1}{N} \\sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)\\phi(p(n)) \\] converge for any polynomial $p$ with real coefficients, and any continuous function $\\phi$ from the torus to the set of complex numbers . We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial $p$, then the avera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0463","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":"2026-05-18T01:34:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5vCT1d0b46cyvCUOtNHGZI64vuSFsbcPzldOPjN4la08ZbM6+u5XnP+NCTzEmAKeQEDWMl9uYt9MkL089m/1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T10:04:06.598952Z"},"content_sha256":"71ce807537b04cf0dd5e294777d54618cb3e5f657a8b0642966f24c8f24d4748","schema_version":"1.0","event_id":"sha256:71ce807537b04cf0dd5e294777d54618cb3e5f657a8b0642966f24c8f24d4748"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT/bundle.json","state_url":"https://pith.science/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT/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":"2026-06-23T10:04:06Z","links":{"resolver":"https://pith.science/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT","bundle":"https://pith.science/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT/bundle.json","state":"https://pith.science/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5PKRM7QQBZ5LSBTM3EGNPYMXHT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5PKRM7QQBZ5LSBTM3EGNPYMXHT","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":"5d2cd16a66ed4ff10ae778104e26a160c2c2f7a9268caafd5c0a3a2c9ad04fff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-25T02:09:18Z","title_canon_sha256":"1fd0c000916fe7979df4afcb0eb37f0078b40b0787d4ef2cd37fc7903cd27e9c"},"schema_version":"1.0","source":{"id":"1409.0463","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0463","created_at":"2026-05-18T01:34:05Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0463v2","created_at":"2026-05-18T01:34:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0463","created_at":"2026-05-18T01:34:05Z"},{"alias_kind":"pith_short_12","alias_value":"5PKRM7QQBZ5L","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5PKRM7QQBZ5LSBTM","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5PKRM7QQ","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:71ce807537b04cf0dd5e294777d54618cb3e5f657a8b0642966f24c8f24d4748","target":"graph","created_at":"2026-05-18T01:34:05Z","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 extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages \\[ \\frac{1}{N} \\sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)\\phi(p(n)) \\] converge for any polynomial $p$ with real coefficients, and any continuous function $\\phi$ from the torus to the set of complex numbers . We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial $p$, then the avera","authors_text":"Idris Assani, Ryo Moore","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-25T02:09:18Z","title":"Extension of Wiener-Wintner double recurrence theorem to polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0463","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":"sha256:0ffe1667ae7724d8687b83186c05f8bb7d071cbcec41556973250036cd00cf7a","target":"record","created_at":"2026-05-18T01:34:05Z","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":"5d2cd16a66ed4ff10ae778104e26a160c2c2f7a9268caafd5c0a3a2c9ad04fff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-25T02:09:18Z","title_canon_sha256":"1fd0c000916fe7979df4afcb0eb37f0078b40b0787d4ef2cd37fc7903cd27e9c"},"schema_version":"1.0","source":{"id":"1409.0463","kind":"arxiv","version":2}},"canonical_sha256":"ebd5167e100e7ab9066cd90cd7e1973cf94b984cf5314019be890cd2379d7e4e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ebd5167e100e7ab9066cd90cd7e1973cf94b984cf5314019be890cd2379d7e4e","first_computed_at":"2026-05-18T01:34:05.573072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:05.573072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UJggvj7YYld8amTytjq/xw928zsw3L1uzp3M1VN0rqqFfsMvoUJP2RS33uxfAVY7SJgfIw/36nbOwyPIvojJAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:05.573691Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.0463","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ffe1667ae7724d8687b83186c05f8bb7d071cbcec41556973250036cd00cf7a","sha256:71ce807537b04cf0dd5e294777d54618cb3e5f657a8b0642966f24c8f24d4748"],"state_sha256":"b978acb79d41b3919d38a6e984262636ef2ddc9129c6f699db81738af17b5cbf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EloTjME5ayyJTK3wmbPrUX77vpMXQSJ2vjmyfpVrvx5RkLmajVUb5fSeqE1QqUw5yCHIIeZUebU5CuIvqwDmDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T10:04:06.600989Z","bundle_sha256":"4b4c355dc164ea56208d29d8a3741dc128ce37da4b8a1dfe356b7eefbeed0027"}}