{"paper":{"title":"Dimension bound for badly approximable grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Nicolas de Saxc\\'e, Seonhee Lim, Uri Shapira","submitted_at":"2017-06-29T07:23:14Z","abstract_excerpt":"We show that for almost any vector $v$ in $\\mathbb{R}^n$, for any $\\epsilon>0$ there exists $\\delta>0$ such that the dimension of the set of vectors $w$ satisfying $\\liminf_{k\\to\\infty} k^{1/n}<kv-w> \\ge \\epsilon$ (where $<\\cdot>$ denotes the distance from the nearest integer), is bounded above by $n-\\delta$. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09600","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":""},"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"}