{"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"}