{"paper":{"title":"Schmidt's game, fractals, and numbers normal to no base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Barak Weiss, Dmitry Kleinbock, Lior Fishman, Ryan Broderick, Yann Bugeaud","submitted_at":"2009-09-23T16:36:13Z","abstract_excerpt":"Given $b > 1$ and $y \\in \\mathbb{R}/\\mathbb{Z}$, we consider the set of $x\\in \\mathbb{R}$ such that $y$ is not a limit point of the sequence $\\{b^n x \\bmod 1: n\\in\\mathbb{N}\\}$. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with `sufficiently regular' fractals $K\\subset \\mathbb{R}$ (that is, supporting measures $\\mu$ satisfying certain decay conditions). Furthermore, the i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4251","kind":"arxiv","version":4},"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"}