{"paper":{"title":"New improvement to Falconer distance set problem in higher dimensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Kevin Ren, Ruixiang Zhang, Xiumin Du, Yumeng Ou","submitted_at":"2023-09-08T03:46:50Z","abstract_excerpt":"We show that if a compact set $E\\subset \\mathbb{R}^d$ has Hausdorff dimension larger than $\\frac{d}{2}+\\frac{1}{4}-\\frac{1}{8d+4}$, where $d\\geq 3$, then there is a point $x\\in E$ such that the pinned distance set $\\Delta_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \\ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\\dim_H (E) \\in (\\frac{d}{2} - \\frac{1}{4} - \\frac{3}{8d+4}, \\frac{d}{2}+\\frac{1}{4}-\\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.04103","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.04103/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"}