{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:T76ZSWDEDIMNI2WSEEITKIFFE4","short_pith_number":"pith:T76ZSWDE","schema_version":"1.0","canonical_sha256":"9ffd9958641a18d46ad221113520a5270f84006cc816d3a25d69eafec5366cfe","source":{"kind":"arxiv","id":"2212.02496","version":1},"attestation_state":"computed","paper":{"title":"Cosine Sign Correlation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Ansel Goh, Gavin Pettigrew, Kevin Liu, Madeline Legate, Shilin Dou","submitted_at":"2022-12-05T18:58:34Z","abstract_excerpt":"Fix $\\left\\{a_1, \\dots, a_n \\right\\} \\subset \\mathbb{N}$, and let $x$ be a uniformly distributed random variable on $[0,2\\pi]$. The probability $\\mathbb{P}(a_1,\\ldots,a_n)$ that $\\cos(a_1 x), \\dots, \\cos(a_n x)$ are either all positive or all negative is non-zero since $\\cos(a_i x) \\sim 1$ for $x$ in a neighborhood of $0$. We are interested in how small this probability can be. Motivated by a problem in spectral theory, Goncalves, Oliveira e Silva, and Steinerberger proved that $\\mathbb{P}(a_1,a_2) \\geq 1/3$ with equality if and only if $\\left\\{a_1, a_2 \\right\\} = \\gcd(a_1, a_2)\\cdot \\left\\{1,"},"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":"2212.02496","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2022-12-05T18:58:34Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ce897d79037fddb211b5524aea55e0284737209a9f6957ae8f99f2bb6c1001df","abstract_canon_sha256":"1cd1a25581c31f2f07b510ee57ce68d4c698579fd19d3f312a5b42f64613feb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:22:24.986457Z","signature_b64":"aBdzFV2/TACgJtcbDmXFgxtoKcvY9AABEk7X7XHsGfTZGt5RyNqc9dPIc08h8QLiB7VQGep0bw9pdQFndpHjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ffd9958641a18d46ad221113520a5270f84006cc816d3a25d69eafec5366cfe","last_reissued_at":"2026-07-05T05:22:24.985962Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:22:24.985962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cosine Sign Correlation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Ansel Goh, Gavin Pettigrew, Kevin Liu, Madeline Legate, Shilin Dou","submitted_at":"2022-12-05T18:58:34Z","abstract_excerpt":"Fix $\\left\\{a_1, \\dots, a_n \\right\\} \\subset \\mathbb{N}$, and let $x$ be a uniformly distributed random variable on $[0,2\\pi]$. The probability $\\mathbb{P}(a_1,\\ldots,a_n)$ that $\\cos(a_1 x), \\dots, \\cos(a_n x)$ are either all positive or all negative is non-zero since $\\cos(a_i x) \\sim 1$ for $x$ in a neighborhood of $0$. We are interested in how small this probability can be. Motivated by a problem in spectral theory, Goncalves, Oliveira e Silva, and Steinerberger proved that $\\mathbb{P}(a_1,a_2) \\geq 1/3$ with equality if and only if $\\left\\{a_1, a_2 \\right\\} = \\gcd(a_1, a_2)\\cdot \\left\\{1,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.02496","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2212.02496/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2212.02496","created_at":"2026-07-05T05:22:24.986024+00:00"},{"alias_kind":"arxiv_version","alias_value":"2212.02496v1","created_at":"2026-07-05T05:22:24.986024+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.02496","created_at":"2026-07-05T05:22:24.986024+00:00"},{"alias_kind":"pith_short_12","alias_value":"T76ZSWDEDIMN","created_at":"2026-07-05T05:22:24.986024+00:00"},{"alias_kind":"pith_short_16","alias_value":"T76ZSWDEDIMNI2WS","created_at":"2026-07-05T05:22:24.986024+00:00"},{"alias_kind":"pith_short_8","alias_value":"T76ZSWDE","created_at":"2026-07-05T05:22:24.986024+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/T76ZSWDEDIMNI2WSEEITKIFFE4","json":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4.json","graph_json":"https://pith.science/api/pith-number/T76ZSWDEDIMNI2WSEEITKIFFE4/graph.json","events_json":"https://pith.science/api/pith-number/T76ZSWDEDIMNI2WSEEITKIFFE4/events.json","paper":"https://pith.science/paper/T76ZSWDE"},"agent_actions":{"view_html":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4","download_json":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4.json","view_paper":"https://pith.science/paper/T76ZSWDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2212.02496&json=true","fetch_graph":"https://pith.science/api/pith-number/T76ZSWDEDIMNI2WSEEITKIFFE4/graph.json","fetch_events":"https://pith.science/api/pith-number/T76ZSWDEDIMNI2WSEEITKIFFE4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4/action/storage_attestation","attest_author":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4/action/author_attestation","sign_citation":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4/action/citation_signature","submit_replication":"https://pith.science/pith/T76ZSWDEDIMNI2WSEEITKIFFE4/action/replication_record"}},"created_at":"2026-07-05T05:22:24.986024+00:00","updated_at":"2026-07-05T05:22:24.986024+00:00"}