{"paper":{"title":"On the typical values of the cross-correlation measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"L\\'aszl\\'o M\\'erai","submitted_at":"2016-03-03T08:28:16Z","abstract_excerpt":"Gyarmati, Mauduit and S\\'ark\\\"ozy introduced the \\textit{cross-correlation measure} $\\Phi_k(\\mathcal{F})$ to measure the randomness of families of binary sequences $\\mathcal{F} \\subset \\{-1,1\\}^N$.\n  In this paper we study the order of magnitude of the cross-correlation measure $\\Phi_k(\\mathcal{F})$ for typical families. We prove that, for most families $\\mathcal{F} \\subset \\{-1,1\\}^N$ of size $2\\leq |\\mathcal{F}|<2^{N/12}$, $\\Phi_k(\\mathcal{F})$ is of order $\\sqrt{N\\log \\binom{N}{k}+k\\log |\\mathcal{F}|}$ for any given $2\\leq k \\leq N/(6\\log_2 |\\mathcal{F}|)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01018","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"}