{"paper":{"title":"Upper and lower densities have the strong Darboux property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.NT"],"primary_cat":"math.CA","authors_text":"Paolo Leonetti, Salvatore Tringali","submitted_at":"2015-10-26T13:25:05Z","abstract_excerpt":"Let $\\mathcal{P}({\\bf N})$ be the power set of $\\bf N$. An upper density (on $\\bf N$) is a non\\-decreasing and subadditive function $\\mu^\\ast: \\mathcal{P}({\\bf N})\\to\\bf R$ such that $\\mu^\\ast({\\bf N}) = 1$ and $\\mu^\\ast(k \\cdot X + h) = \\frac{1}{k} \\mu^\\ast(X)$ for all $X \\subseteq \\bf N$ and $h,k \\in {\\bf N}^+$, where $k \\cdot X + h := \\{kx + h: x \\in X\\}$.\n  The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper P\\'olya, and upper analytic densities are examples of upper densities.\n  We show that every upper density $\\mu^\\ast$ has the strong Darboux property, and so does t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07473","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"}