{"paper":{"title":"Tangent points of lower content $d$-regular sets and $\\beta$ numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Michele Villa","submitted_at":"2017-12-07T19:26:50Z","abstract_excerpt":"Given a lower content $d$-regular set in $\\mathbb{R}^n$, we prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones $\\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $\\mathcal{H}^d$-measure zero. The main point of our result is that $\\mathcal{H}^d|_E$ is not assumed to be $\\sigma$-finite; because of this, we use a certain variant of the $\\beta$ coefficient, firstly introduced by Azzam and Schul in [AS1], which is given in terms of integration with respect to the Hausdorff content."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02823","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1712.02823/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}