{"paper":{"title":"Denjoy, Demuth, and Density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Andre Nies, Joseph S. Miller, Laurent Bienvenu, Rupert H\\\"olzl","submitted_at":"2013-08-29T08:47:54Z","abstract_excerpt":"We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy-Young-Saks theorem. For the first, we show that a Martin-Loef random real $z\\in [0,1]$ is Turing incomplete if and only if every effectively closed class $C \\subseteq [0,1]$ containing $z$ has positive density at $z$. Under the stronger assumption that $z$ is not LR-hard, we show that $z$ has density-one in every such class. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Loef random reals and $K$-trivial sets: the non-cu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.6402","kind":"arxiv","version":3},"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"}