{"paper":{"title":"Analytic sets of reals and the density function in the Cantor space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alessandro Andretta, Riccardo Camerlo","submitted_at":"2017-05-05T16:20:24Z","abstract_excerpt":"We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\\mathcal{U}$ for $\\Sigma^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\\mathcal{U}$ is the set of all pairs $( K , r )$ with $K$ compact and $r \\in ( 0 ; 1 )$ being the density of some point with respect to $K$. This result yields that the set of all $K$ such that the range of its density function is $S \\cup \\{ 0 , 1 \\}$, for some fixed uncountable analytic set $S \\subseteq ( 0 ; 1 )$, is $\\Pi^{1}_{2}$-complete."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02285","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"}