{"paper":{"title":"Badly approximable points on planar curves and winning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinpeng An, Sanju Velani, Victor Beresnevich","submitted_at":"2014-08-29T23:59:55Z","abstract_excerpt":"For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) \\in R^2 such that max \\{ ||qx||^{1/i}, \\, ||qy||^{1/j} \\} > c/q for some positive constant c = c(x,y) and all q in N. We show that \\Bad(i,j) \\cap C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0064","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"}