{"paper":{"title":"Hausdorff dimension in inhomogeneous Diophantine approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Dong Han Kim, Micha{\\l} Rams, Seonhee Lim, Yann Bugeaud","submitted_at":"2018-05-26T06:56:48Z","abstract_excerpt":"Let $\\alpha$ be an irrational real number. We show that the set of $\\epsilon$-badly approximable numbers \\[ \\mathrm{Bad}^\\varepsilon (\\alpha) := \\{x\\in [0,1]\\, : \\, \\liminf_{|q| \\to \\infty} |q| \\cdot \\| q\\alpha -x \\| \\geq \\varepsilon \\} \\] has full Hausdorff dimension for some positive $\\epsilon$ if and only if $\\alpha$ is singular on average. The condition is equivalent to the average $\\frac{1}{k} \\sum_{i=1, \\cdots, k} \\log a_i$ of the logarithms of the partial quotients $a_i$ of $\\alpha$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10436","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"}