{"paper":{"title":"Pair Correlation for Fractional Parts of $\\alpha n^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"D.R. Heath-Brown","submitted_at":"2009-04-04T14:21:31Z","abstract_excerpt":"We construct real numbers $\\alpha$ for which the pair correlation function \\[N^{-1}#\\{m<n\\le N:||\\alpha m^2-\\alpha n^2||\\le XN^{-1}\\}\\] tends to $X$ as $N$ grows. Moreover we show for any \"Diophantine\" $\\alpha$ that the pair correlation function is $X+O(X^{7/8})+O((\\log N)^{-1}$ for $1\\le X\\le\\log N$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.0714","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"}