{"paper":{"title":"Dimension estimates for badly approximable affine forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Seonhee Lim, Taehyeong Kim, Wooyeon Kim","submitted_at":"2021-11-30T14:03:57Z","abstract_excerpt":"For given $\\epsilon>0$ and $b\\in\\mathbb{R}^m$, we say that a real $m\\times n$ matrix $A$ is $\\epsilon$-badly approximable for the target $b$ if $$\\liminf_{q\\in\\mathbb{Z}^n, \\|q\\|\\to\\infty} \\|q\\|^n \\langle Aq-b \\rangle^m \\geq \\epsilon,$$ where $\\langle \\cdot \\rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $\\epsilon$-badly approximable matrices for fixed target $b$ and the set of $\\epsilon$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.15410","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/2111.15410/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"}