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(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"},"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":"0910.5412","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-10-28T16:07:14Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e5fd5f9ee21e3a8ea46f91736eebf5aca644319c4d01da4fc6d7f9c67e5c873d","abstract_canon_sha256":"50de60ce3f328ab32b5321b90037800cb370c763e0866d1daef7eadd10ad5dfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:22.592106Z","signature_b64":"sXv37DWDSIACjISp52GKNI2SjN8CbQUHbOXI9n8yi56+CL67trWwGWu5kKV3YubJL9HxajPTyB9NwUPEUcd7BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d703a47e0d61810e78a78d35552079ef445f6f7d878132d08a50c65c212988a","last_reissued_at":"2026-05-18T03:49:22.591177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:22.591177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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