{"paper":{"title":"Hausdorff dimension of unions of $k$-planes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Shengwen Gan","submitted_at":"2023-05-23T22:00:16Z","abstract_excerpt":"We prove a conjecture of H\\'era on the dimension of unions of $k$-planes. Let $0<k \\le d<n$ be integers, and $\\beta\\in[0,k+1)$. If $\\mathcal{V}\\subset A(k,n)$, with $\\text{dim}(\\mathcal{V})=(k+1)(d-k)+\\beta$, then $\\text{dim}(\\bigcup_{V\\in\\mathcal{V}}V)\\ge d+\\min\\{1,\\beta\\}$. The proof combines a recent idea of Zahl and the Brascamp-Lieb inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.14544","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/2305.14544/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"}