{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:VHF2ZZLQQKBQ5D5XRYQHAOIZRS","short_pith_number":"pith:VHF2ZZLQ","schema_version":"1.0","canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","source":{"kind":"arxiv","id":"1204.3441","version":1},"attestation_state":"computed","paper":{"title":"Sharp geometric rigidity of isometries on Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"D. V. Isangulova, S. K. Vodopyanov","submitted_at":"2012-04-16T11:15:25Z","abstract_excerpt":"We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\\sqrt{\\varepsilon}+\\varepsilon$ in the uniform norm, and up to proximity order $\\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results."},"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":"1204.3441","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-04-16T11:15:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"118c11ff436385f0e235480925bb39d93cfcbb635f07bfc09f00e9296aace525","abstract_canon_sha256":"df2b89c6c0a9bd98021d52dae7f01c3750d3a50d08ac34f3ae343aa0a80fd26b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:51.530695Z","signature_b64":"04dx4/jrW9Ro8DbRMLl+YB+s2rk+8tFP7+U1bJ+ZsZLGMYhwnYUwI4JoCRAx42TobFaWI0QdDFD/VeSLdlMgDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","last_reissued_at":"2026-05-18T03:57:51.530069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:51.530069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp geometric rigidity of isometries on Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"D. V. Isangulova, S. K. Vodopyanov","submitted_at":"2012-04-16T11:15:25Z","abstract_excerpt":"We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\\sqrt{\\varepsilon}+\\varepsilon$ in the uniform norm, and up to proximity order $\\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3441","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":"1204.3441","created_at":"2026-05-18T03:57:51.530159+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.3441v1","created_at":"2026-05-18T03:57:51.530159+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3441","created_at":"2026-05-18T03:57:51.530159+00:00"},{"alias_kind":"pith_short_12","alias_value":"VHF2ZZLQQKBQ","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VHF2ZZLQQKBQ5D5X","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VHF2ZZLQ","created_at":"2026-05-18T12:27:25.539911+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/VHF2ZZLQQKBQ5D5XRYQHAOIZRS","json":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS.json","graph_json":"https://pith.science/api/pith-number/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/graph.json","events_json":"https://pith.science/api/pith-number/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/events.json","paper":"https://pith.science/paper/VHF2ZZLQ"},"agent_actions":{"view_html":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS","download_json":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS.json","view_paper":"https://pith.science/paper/VHF2ZZLQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.3441&json=true","fetch_graph":"https://pith.science/api/pith-number/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/graph.json","fetch_events":"https://pith.science/api/pith-number/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/action/storage_attestation","attest_author":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/action/author_attestation","sign_citation":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/action/citation_signature","submit_replication":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/action/replication_record"}},"created_at":"2026-05-18T03:57:51.530159+00:00","updated_at":"2026-05-18T03:57:51.530159+00:00"}