{"paper":{"title":"Rectifiability via a square function and Preiss' theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Tatiana Toro, Xavier Tolsa","submitted_at":"2014-02-12T12:18:16Z","abstract_excerpt":"Let $E$ be a set in $\\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\\liminf_{r\\to0}r^{-n} H^n(B(x,r)\\cap E)>0$ for $H^n$-a.e. $x\\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $$\\int_0^1 \\left|\\frac{H^n(B(x,r)\\cap E)}{r^n} - \\frac{H^n(B(x,2r)\\cap E)}{(2r)^n}\\right|^2\\,\\frac{dr}r < \\infty$$ for $H^n$-a.e. $x\\in E$; and also if and only if $$ \\lim_{r\\to0}\\left(\\frac{H^n(B(x,r)\\cap E)}{r^n} - \\frac{H^n(B(x,2r)\\cap E)}{(2r)^n}\\right) = 0$$ for $H^n$-a.e. $x\\in E$. Other more general results involving Radon measures are also proved."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2799","kind":"arxiv","version":4},"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"}