{"paper":{"title":"Borel-Cantelli sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Jon Chaika, Michael Boshernitzan","submitted_at":"2009-10-28T16:07:14Z","abstract_excerpt":"A sequence $\\{x_{n}\\}_1^\\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\\{a_n\\}$ with $\\underset{i=1}{\\overset{\\infty}{\\sum}}a_i=\\infty$ the set \\[\\underset{k=1}{\\overset{\\infty}{\\cap}} \\underset{n=k}{\\overset{\\infty}{\\cup}} B(x_n, a_n))=\\{x\\in[0,1)\\mid |x_n-x|<a_n \\text{for} \\infty \\text{many}n\\geq1\\}\\] has full Lebesgue measure. (To put it informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds).\n  The notion of BC sequences is motivated by the Monotone Shrinking Target Property for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5412","kind":"arxiv","version":2},"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"}