{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:Z2LHWERHNVZ7G4NUSSFFM6ZYXL","short_pith_number":"pith:Z2LHWERH","schema_version":"1.0","canonical_sha256":"ce967b12276d73f371b4948a567b38bae607a0428311a2f0fcdc4adb120bd5af","source":{"kind":"arxiv","id":"1507.06826","version":1},"attestation_state":"computed","paper":{"title":"Classifying $\\mathsf{GL}(n,\\mathbb Z)$-orbits of points and rational subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici, Leonardo Manuel Cabrer","submitted_at":"2015-07-24T12:52:26Z","abstract_excerpt":"We first show that the subgroup of the abelian real group $\\mathbb{R}$ generated by the coordinates of a point in $x = (x_1,\\dots,x_n)\\in\\mathbb{R}^n$ completely classifies the $\\mathsf{GL}(n,\\mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Dani's theorem: the $\\mathsf{GL}(n,\\mathbb Z)$-orbit of $x\\in\\mathbb{R}^n$ is dense iff $x_i/x_j\\in \\mathbb{R} \\setminus \\mathbb Q$ for some $i,j=1,\\dots,n$. We then classify $\\mathsf{GL}(n,\\mathbb Z)$-orbits of rational affine subspaces $F$ of $\\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope a"},"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":"1507.06826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-24T12:52:26Z","cross_cats_sorted":[],"title_canon_sha256":"3bff71653dc2cae399591b19019048bd7f0803a9f8490f92082ce859cf68ff76","abstract_canon_sha256":"597bcbbae8ab45322975889d1f025597b5ad49ac8de7d9c0b02e37c652f361e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:21.935914Z","signature_b64":"xOQ+7U08lEE1ZV4jOG2sOBFHR5O9PaodjMfxM8dZKc2C1sJrmPAN4ydmwo6IaHT4n1AaWmCrSsXbsmpL2CSwDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce967b12276d73f371b4948a567b38bae607a0428311a2f0fcdc4adb120bd5af","last_reissued_at":"2026-05-18T01:36:21.935406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:21.935406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classifying $\\mathsf{GL}(n,\\mathbb Z)$-orbits of points and rational subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici, Leonardo Manuel Cabrer","submitted_at":"2015-07-24T12:52:26Z","abstract_excerpt":"We first show that the subgroup of the abelian real group $\\mathbb{R}$ generated by the coordinates of a point in $x = (x_1,\\dots,x_n)\\in\\mathbb{R}^n$ completely classifies the $\\mathsf{GL}(n,\\mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Dani's theorem: the $\\mathsf{GL}(n,\\mathbb Z)$-orbit of $x\\in\\mathbb{R}^n$ is dense iff $x_i/x_j\\in \\mathbb{R} \\setminus \\mathbb Q$ for some $i,j=1,\\dots,n$. We then classify $\\mathsf{GL}(n,\\mathbb Z)$-orbits of rational affine subspaces $F$ of $\\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06826","kind":"arxiv","version":1},"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":"1507.06826","created_at":"2026-05-18T01:36:21.935504+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.06826v1","created_at":"2026-05-18T01:36:21.935504+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06826","created_at":"2026-05-18T01:36:21.935504+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z2LHWERHNVZ7","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z2LHWERHNVZ7G4NU","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z2LHWERH","created_at":"2026-05-18T12:29:52.810259+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/Z2LHWERHNVZ7G4NUSSFFM6ZYXL","json":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL.json","graph_json":"https://pith.science/api/pith-number/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/graph.json","events_json":"https://pith.science/api/pith-number/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/events.json","paper":"https://pith.science/paper/Z2LHWERH"},"agent_actions":{"view_html":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL","download_json":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL.json","view_paper":"https://pith.science/paper/Z2LHWERH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.06826&json=true","fetch_graph":"https://pith.science/api/pith-number/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/graph.json","fetch_events":"https://pith.science/api/pith-number/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/action/storage_attestation","attest_author":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/action/author_attestation","sign_citation":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/action/citation_signature","submit_replication":"https://pith.science/pith/Z2LHWERHNVZ7G4NUSSFFM6ZYXL/action/replication_record"}},"created_at":"2026-05-18T01:36:21.935504+00:00","updated_at":"2026-05-18T01:36:21.935504+00:00"}