{"paper":{"title":"Schmidt's game, fractals, and orbits of toral endomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Dmitry Kleinbock, Lior Fishman, Ryan Broderick","submitted_at":"2010-01-02T16:40:43Z","abstract_excerpt":"Given an integer nonsingular $n\\times n$ matrix $M$ and a point $y \\in \\mathbb{R}^n/\\mathbb{Z}^n$, consider the set $\\tilde E(M,y)$ of vectors $x\\in \\mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\\{M^k x \\mod \\mathbb{Z}^n: k\\in\\mathbb{N}\\}$. S.G. Dani showed in 1988 that whenever $M$ is semisimple and $y \\in \\mathbb{Q}^n/\\mathbb{Z}^n$, the set $\\tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary $y \\in \\mathbb{R}^n/\\mathbb{Z}^n$ and integer nonsingular $M$, and in fact replacing the sequence of powers of $M$ by any "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0318","kind":"arxiv","version":3},"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"}