{"paper":{"title":"On the speed of convergence in the strong density theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Constantinos Gryllakis (Department of Mathematics, Kapodistrian University of Athens), National, Panagiotis Georgopoulos","submitted_at":"2018-05-25T08:37:02Z","abstract_excerpt":"For a compact set $K\\subset \\mathbb{R}^m$, we have two indexes given under simple parameters of the set $K$ (these parameters go back to Besicovitch and Taylor in the late 50's). In the present paper we prove that with the exception of a single extreme value for each index, we have the following elementary estimate on how fast the ratio in the strong density theorem of Saks will tend to one \\[ \\frac{|R\\cap K|}{|R|}>1-o\\bigg(\\frac{1}{|\\log d(R)|}\\bigg) \\qquad \\text{for a.e.} \\ \\ x\\in K \\ \\ \\text{and for} \\ \\ d(R)\\to 0 \\] (provided $x\\in R$, where $R$ is an interval in $\\mathbb{R}^m$, $d$ stands"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10035","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"}