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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1001.0318","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-02T16:40:43Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"4bba00a303cd87c1086a1aa66789b36a9c444a994c064a6a7a58cd748d22986b","abstract_canon_sha256":"e9bc4a90b424dff3ccb7d541f5752294fc55969be46317f9c47e9be6f013a20b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:15.127981Z","signature_b64":"77tTJ6tR/RcjM2oDRDYaopbu5VzannirxH+q6S7UjGBXiQeeS/b7UwPVUesygOwYSqxTOg4TspomXusghHoPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"558eec21c7d15873c715090b5202ebccb92a2ef7c2b602e28759ff8364768d43","last_reissued_at":"2026-05-18T00:05:15.127358Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:15.127358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\{M^k x \\mod \\mathbb{Z}^n: k\\in\\mathbb{N}\\}$. 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