{"paper":{"title":"Hausdorff dimension of restricted Kakeya sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Jonathan M. Fraser, Lijian Yang","submitted_at":"2025-05-09T01:05:18Z","abstract_excerpt":"A Kakeya set in $\\mathbb{R}^n$ is a compact set that contains a unit line segment $I_e$ in each direction $e \\in S^{n-1}$. The Kakeya conjecture states that any Kakeya set in $\\mathbb{R}^n$ has Hausdorff dimension $n$. We consider a restricted case where the midpoint of each line segment $I_e$ must belong to a fixed set $A$ with packing dimension at most $s \\in [0, n]$. In this case, we show that the Hausdorff dimension of the Kakeya set is at least $n - s$. Furthermore, using the \"bush argument\", we improve the lower bound to $\\max \\{ n - s, n - g_n(s)\\}$, where $g_n(s)$ is defined inductivel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.05709","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/2505.05709/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"}