{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:AR4JBYD5LXT6MACMU2B3MXNP7S","short_pith_number":"pith:AR4JBYD5","schema_version":"1.0","canonical_sha256":"047890e07d5de7e6004ca683b65daffcbfd1472dc83cdd810baeb6d17f60f085","source":{"kind":"arxiv","id":"1505.06064","version":1},"attestation_state":"computed","paper":{"title":"A zero-sqrt(5)/ 2 law for cosine families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jean Esterle (IMB)","submitted_at":"2015-05-22T13:24:44Z","abstract_excerpt":"Let $a \\in \\R,$ and let $k(a)$ be the largest constant such that $sup\\vert cos(na)-cos(nb)\\vert \\textless{} k(a)$ for $b\\in \\R$ implies that $b \\in \\pm a+2\\pi\\Z. $ We show that\nif a cosine sequence $(C(n))\\_{n\\in \\Z}$ with values in a Banach algebra $A$ satisfies $sup\\_{n\\ge 1}\\Vert C(n) -cos(na).1\\_A\\Vert \\textless{} k(a),$ then $C(n)=cos(na)$ for $n\\in \\Z.$ Since\n${\\sqrt 5\\over 2} \\le k(a) \\le {8\\over 3\\sqrt 3}$ for every $a \\in \\R,$ this shows that if some cosine family $(C(g))\\_{g\\in G}$ over an abelian group $G$ in a Banach algebra satisfies $sup\\_{g\\in G}\\Vert C(g)-c(g)\\Vert \\textless{} "},"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":"1505.06064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-22T13:24:44Z","cross_cats_sorted":[],"title_canon_sha256":"abc6b398db7a9cb485978cb5cbc15cb0673e65908236aef77ae54a70e446035d","abstract_canon_sha256":"e9ae46a52f89401c13ef009dd6b590d245162087be6844ea861a92a2d1ebe2d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:49.250852Z","signature_b64":"y6MvFjwk7FsaLtP/NAqKEFDSyg9FZL0HMKqW/QfxxJRcBWjyw8247a+tR17gbjzZvw1R2Kvhl4Dc9K4igFriDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"047890e07d5de7e6004ca683b65daffcbfd1472dc83cdd810baeb6d17f60f085","last_reissued_at":"2026-05-18T02:03:49.250094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:49.250094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A zero-sqrt(5)/ 2 law for cosine families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jean Esterle (IMB)","submitted_at":"2015-05-22T13:24:44Z","abstract_excerpt":"Let $a \\in \\R,$ and let $k(a)$ be the largest constant such that $sup\\vert cos(na)-cos(nb)\\vert \\textless{} k(a)$ for $b\\in \\R$ implies that $b \\in \\pm a+2\\pi\\Z. $ We show that\nif a cosine sequence $(C(n))\\_{n\\in \\Z}$ with values in a Banach algebra $A$ satisfies $sup\\_{n\\ge 1}\\Vert C(n) -cos(na).1\\_A\\Vert \\textless{} k(a),$ then $C(n)=cos(na)$ for $n\\in \\Z.$ Since\n${\\sqrt 5\\over 2} \\le k(a) \\le {8\\over 3\\sqrt 3}$ for every $a \\in \\R,$ this shows that if some cosine family $(C(g))\\_{g\\in G}$ over an abelian group $G$ in a Banach algebra satisfies $sup\\_{g\\in G}\\Vert C(g)-c(g)\\Vert \\textless{} "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06064","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.06064","created_at":"2026-05-18T02:03:49.250243+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.06064v1","created_at":"2026-05-18T02:03:49.250243+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06064","created_at":"2026-05-18T02:03:49.250243+00:00"},{"alias_kind":"pith_short_12","alias_value":"AR4JBYD5LXT6","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"AR4JBYD5LXT6MACM","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"AR4JBYD5","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S","json":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S.json","graph_json":"https://pith.science/api/pith-number/AR4JBYD5LXT6MACMU2B3MXNP7S/graph.json","events_json":"https://pith.science/api/pith-number/AR4JBYD5LXT6MACMU2B3MXNP7S/events.json","paper":"https://pith.science/paper/AR4JBYD5"},"agent_actions":{"view_html":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S","download_json":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S.json","view_paper":"https://pith.science/paper/AR4JBYD5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.06064&json=true","fetch_graph":"https://pith.science/api/pith-number/AR4JBYD5LXT6MACMU2B3MXNP7S/graph.json","fetch_events":"https://pith.science/api/pith-number/AR4JBYD5LXT6MACMU2B3MXNP7S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S/action/storage_attestation","attest_author":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S/action/author_attestation","sign_citation":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S/action/citation_signature","submit_replication":"https://pith.science/pith/AR4JBYD5LXT6MACMU2B3MXNP7S/action/replication_record"}},"created_at":"2026-05-18T02:03:49.250243+00:00","updated_at":"2026-05-18T02:03:49.250243+00:00"}