{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:KWHOYIOH2FMHHRYVBEFVEAXLZS","short_pith_number":"pith:KWHOYIOH","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"},"canonical_sha256":"558eec21c7d15873c715090b5202ebccb92a2ef7c2b602e28759ff8364768d43","source":{"kind":"arxiv","id":"1001.0318","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.0318","created_at":"2026-05-18T00:05:15Z"},{"alias_kind":"arxiv_version","alias_value":"1001.0318v3","created_at":"2026-05-18T00:05:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0318","created_at":"2026-05-18T00:05:15Z"},{"alias_kind":"pith_short_12","alias_value":"KWHOYIOH2FMH","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KWHOYIOH2FMHHRYV","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KWHOYIOH","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:KWHOYIOH2FMHHRYVBEFVEAXLZS","target":"record","payload":{"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"},"canonical_sha256":"558eec21c7d15873c715090b5202ebccb92a2ef7c2b602e28759ff8364768d43","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"},"source_kind":"arxiv","source_id":"1001.0318","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:05:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oj1vI74AOUxFhdeNUt1ho7zehMDZX17a+YAum0ULddbmJlKpZcH1PD7rTkWepj0I0DuemhQZ6YMrb66e+nFTCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T20:16:31.260265Z"},"content_sha256":"b0846fcb04a0e434349e9d52f48813e3fd3636ec22a54a92e172c7ce8b337057","schema_version":"1.0","event_id":"sha256:b0846fcb04a0e434349e9d52f48813e3fd3636ec22a54a92e172c7ce8b337057"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:KWHOYIOH2FMHHRYVBEFVEAXLZS","target":"graph","payload":{"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 $\\{M^k x \\mod \\mathbb{Z}^n: k\\in\\mathbb{N}\\}$. 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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:05:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S9k1IjclBaf6jZuhwjvhP7FSYy0FSNmVb7yhsAEcsiOQjM+HcYK2QzwU9BLniv8oFNbkg5aQu2j4UUc1xEQiBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T20:16:31.260634Z"},"content_sha256":"28d2bd207f3cb139c6865eab85193d6d41cb789909e3ffca40616514c7a13bf5","schema_version":"1.0","event_id":"sha256:28d2bd207f3cb139c6865eab85193d6d41cb789909e3ffca40616514c7a13bf5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS/bundle.json","state_url":"https://pith.science/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T20:16:31Z","links":{"resolver":"https://pith.science/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS","bundle":"https://pith.science/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS/bundle.json","state":"https://pith.science/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KWHOYIOH2FMHHRYVBEFVEAXLZS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:KWHOYIOH2FMHHRYVBEFVEAXLZS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e9bc4a90b424dff3ccb7d541f5752294fc55969be46317f9c47e9be6f013a20b","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-02T16:40:43Z","title_canon_sha256":"4bba00a303cd87c1086a1aa66789b36a9c444a994c064a6a7a58cd748d22986b"},"schema_version":"1.0","source":{"id":"1001.0318","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.0318","created_at":"2026-05-18T00:05:15Z"},{"alias_kind":"arxiv_version","alias_value":"1001.0318v3","created_at":"2026-05-18T00:05:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0318","created_at":"2026-05-18T00:05:15Z"},{"alias_kind":"pith_short_12","alias_value":"KWHOYIOH2FMH","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KWHOYIOH2FMHHRYV","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KWHOYIOH","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:28d2bd207f3cb139c6865eab85193d6d41cb789909e3ffca40616514c7a13bf5","target":"graph","created_at":"2026-05-18T00:05:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Dmitry Kleinbock, Lior Fishman, Ryan Broderick","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-02T16:40:43Z","title":"Schmidt's game, fractals, and orbits of toral endomorphisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0318","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b0846fcb04a0e434349e9d52f48813e3fd3636ec22a54a92e172c7ce8b337057","target":"record","created_at":"2026-05-18T00:05:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e9bc4a90b424dff3ccb7d541f5752294fc55969be46317f9c47e9be6f013a20b","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-02T16:40:43Z","title_canon_sha256":"4bba00a303cd87c1086a1aa66789b36a9c444a994c064a6a7a58cd748d22986b"},"schema_version":"1.0","source":{"id":"1001.0318","kind":"arxiv","version":3}},"canonical_sha256":"558eec21c7d15873c715090b5202ebccb92a2ef7c2b602e28759ff8364768d43","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"558eec21c7d15873c715090b5202ebccb92a2ef7c2b602e28759ff8364768d43","first_computed_at":"2026-05-18T00:05:15.127358Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:15.127358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"77tTJ6tR/RcjM2oDRDYaopbu5VzannirxH+q6S7UjGBXiQeeS/b7UwPVUesygOwYSqxTOg4TspomXusghHoPDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:15.127981Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.0318","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0846fcb04a0e434349e9d52f48813e3fd3636ec22a54a92e172c7ce8b337057","sha256:28d2bd207f3cb139c6865eab85193d6d41cb789909e3ffca40616514c7a13bf5"],"state_sha256":"258ca874206bae793264de88508ca0b43a91e2eba0e6b22dbd3f6f2531f7cf4f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zEOTvH0sHBJgnZJHpqCK+BkC6vHcm0m4+tx+fi/KBmKhWQ7iQ2QA+pm4Mgg150k58RBnFpZOMqGXgBQNOJtsCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T20:16:31.262710Z","bundle_sha256":"b9287809f7b3aab598dc9ac8ec35d07b799490a1aa588b14759e922d6389e59b"}}