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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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. 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