{"paper":{"title":"The Assouad dimension of Kakeya sets in $\\mathbb{R}^3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hong Wang, Joshua Zahl","submitted_at":"2024-01-22T20:06:05Z","abstract_excerpt":"This paper studies the structure of Kakeya sets in $\\mathbb{R}^3$. We show that for every Kakeya set $K\\subset\\mathbb{R}^3$, there exist well-separated scales $0<\\delta<\\rho\\leq 1$ so that the $\\delta$ neighborhood of $K$ is almost as large as the $\\rho$ neighborhood of $K$. As a consequence, every Kakeya set in $\\mathbb{R}^3$ has Assouad dimension 3 and every Ahlfors-David regular Kakeya set in $\\mathbb{R}^3$ has Hausdorff dimension 3. We also show that every Kakeya set in $\\mathbb{R}^3$ that has \"stably equal\" Hausdorff and packing dimension (this is a new notion, which is introduced to avoi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.12337","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/2401.12337/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"}