{"paper":{"title":"Hausdorff dimension and exact approximation order in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Nicolas de Saxc\\'e, Prasuna Bandi","submitted_at":"2023-12-15T23:16:42Z","abstract_excerpt":"Given a non-increasing function $\\psi\\colon\\mathbb{N}\\to\\mathbb{R}^+$ such that $s^{\\frac{n+1}{n}}\\psi(s)$ tends to zero as $s$ goes to infinity, we show that the set of points in $\\mathbb{R}^n$ that are exactly $\\psi$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\\geq 2$, this answers questions of Jarn\\'{i}k and of Beresnevich, Dickinson, and Velani."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.10255","kind":"arxiv","version":1},"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/2312.10255/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"}