{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:MVVEXHMHROARE3IMEKCBE7TSBW","short_pith_number":"pith:MVVEXHMH","schema_version":"1.0","canonical_sha256":"656a4b9d878b81126d0c2284127e720d991e46a522d8099c37d1b9bcec4d8f9f","source":{"kind":"arxiv","id":"1809.07404","version":2},"attestation_state":"computed","paper":{"title":"Examples of badly approximable vectors over number fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Robert Hines","submitted_at":"2018-09-19T20:45:34Z","abstract_excerpt":"We consider approximation of vectors $\\mathbf{z}\\in F\\otimes\\mathbb{R}\\cong\\mathbb{R}^r\\times\\mathbb{C}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of $SL_2(\\mathcal{O}_F)\\backslash SL_2(F\\otimes\\mathbb{R})$ naturally associated to (totally indefinite, anisotropic) $F$-rational binary quadratic and Hermitian forms, a generalization of the well-known fact that quadratic irrationals are badly approximable over $\\mathbb{Q}$."},"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":"1809.07404","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-19T20:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"d83ed21bb9e9f0ad67927fe4ef6ba45dcb5ca33a67f46a3b06d0013bcc6fc461","abstract_canon_sha256":"cba44420c0e18256f4d7dbb84a12d62dbf749a1a475157b23fad90d321e56606"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:23.116548Z","signature_b64":"ryWWtmpTI2WJNAW70+LzB7H9EmL1lWQUQ9o+AgRZEocIwwxwGcOQ4rWpS7fSizWqaoPJ5DOg7+WikGovD0YyCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"656a4b9d878b81126d0c2284127e720d991e46a522d8099c37d1b9bcec4d8f9f","last_reissued_at":"2026-05-17T23:56:23.115967Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:23.115967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Examples of badly approximable vectors over number fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Robert Hines","submitted_at":"2018-09-19T20:45:34Z","abstract_excerpt":"We consider approximation of vectors $\\mathbf{z}\\in F\\otimes\\mathbb{R}\\cong\\mathbb{R}^r\\times\\mathbb{C}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of $SL_2(\\mathcal{O}_F)\\backslash SL_2(F\\otimes\\mathbb{R})$ naturally associated to (totally indefinite, anisotropic) $F$-rational binary quadratic and Hermitian forms, a generalization of the well-known fact that quadratic irrationals are badly approximable over $\\mathbb{Q}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07404","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.07404","created_at":"2026-05-17T23:56:23.116063+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.07404v2","created_at":"2026-05-17T23:56:23.116063+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07404","created_at":"2026-05-17T23:56:23.116063+00:00"},{"alias_kind":"pith_short_12","alias_value":"MVVEXHMHROAR","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"MVVEXHMHROARE3IM","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"MVVEXHMH","created_at":"2026-05-18T12:32:40.477152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW","json":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW.json","graph_json":"https://pith.science/api/pith-number/MVVEXHMHROARE3IMEKCBE7TSBW/graph.json","events_json":"https://pith.science/api/pith-number/MVVEXHMHROARE3IMEKCBE7TSBW/events.json","paper":"https://pith.science/paper/MVVEXHMH"},"agent_actions":{"view_html":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW","download_json":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW.json","view_paper":"https://pith.science/paper/MVVEXHMH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.07404&json=true","fetch_graph":"https://pith.science/api/pith-number/MVVEXHMHROARE3IMEKCBE7TSBW/graph.json","fetch_events":"https://pith.science/api/pith-number/MVVEXHMHROARE3IMEKCBE7TSBW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW/action/storage_attestation","attest_author":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW/action/author_attestation","sign_citation":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW/action/citation_signature","submit_replication":"https://pith.science/pith/MVVEXHMHROARE3IMEKCBE7TSBW/action/replication_record"}},"created_at":"2026-05-17T23:56:23.116063+00:00","updated_at":"2026-05-17T23:56:23.116063+00:00"}