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Recently, Gowers and Mili\\'cevi\\'c applied a bilinear Bogolyubov-Ruzsa lemma as part of a proof of the inverse $U^4$ theorem with effective bounds. The goal of this note is to obtain quantitative bounds for the bilinear Bogolyubov-Ruzsa lemma which are similar to those obtained by Sanders for the Bogolyubov-Ruzsa lemma.\n  We show that if a set $A \\subset \\mathbb{F}_p^n \\times \\mathbb{F}_p^n$ has density $\\a"},"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":"1808.04965","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2018-08-15T04:20:22Z","cross_cats_sorted":[],"title_canon_sha256":"61a2dd606adbc97304dc166f9ef140f4ea7baa1bfc2b96b681a3bd507dd563cf","abstract_canon_sha256":"ce8c82a7694216fe7b36662c969c640bc652d7e9e2bda9e680da9652fd00b52a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:23.721556Z","signature_b64":"J/AyvwH33WgqCKQ+0c09V6v4Pc2LtUqC2/D/LHsMM51dIDTLdbanUPsYMsaFbw5jKUint2oN2wFHNKF2O4i7Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a322576fbb503896e3cfd20b0b1facfc9fc0696bb301a30dec42b85f12d8e06","last_reissued_at":"2026-05-17T23:43:23.720839Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:23.720839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kaave Hosseini, Shachar Lovett","submitted_at":"2018-08-15T04:20:22Z","abstract_excerpt":"The Bogolyubov-Ruzsa lemma, in particular the quantitative bounds obtained by Sanders, plays a central role in obtaining effective bounds for the inverse $U^3$ theorem for the Gowers norms. 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