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For the Bohnenblust-Hille inequality and real scalars it is well-known that the optimal constants are not contractive. In this note, among other results, we show that if we consider sums over $M:=M(m)$ indexes with $M\\log M=o(m)$, the optimal constants are contractive. For instance, we can consider% \\[ M=\\left\\lfloor \\frac{m}{\\left( \\log m\\right) ^{1+\\frac{1}{\\log\\log\\log m}}% }\\right\\rfloor \\] where $\\lfloor x\\rfloor:=\\max\\{n\\"},"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":"1705.06307","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-05-17T18:39:38Z","cross_cats_sorted":[],"title_canon_sha256":"ccab61edd71e96a374ddf520dd381527d93162a946e68751719bfc4bdbd37f20","abstract_canon_sha256":"df2b17cac18b19d7ea539a5e4cae09f9175ffbd42cbdfb621b1390f96b6a3654"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:47.548805Z","signature_b64":"YVQLhfEqtPSTQ5qAeuQBcN1hglMGekyPPBkS7SkYZvNVHan3liyTrrSX34qItrVp9YS9eG1HHAZdcbSkGNjuDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c22987d7f55463ff611a4a411c2fa9ba4604cc3d2fd50ab4a06ecb22b55b571e","last_reissued_at":"2026-05-18T00:19:47.548109Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:47.548109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"When are the Hardy-Littlewood inequalities contractive?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"D.M. 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