{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:7WNDKE237NYOPH5LTT7H6DWR2B","short_pith_number":"pith:7WNDKE23","schema_version":"1.0","canonical_sha256":"fd9a35135bfb70e79fab9cfe7f0ed1d05c01a5163a0772d0c45963b0f8331b30","source":{"kind":"arxiv","id":"1309.6223","version":1},"attestation_state":"computed","paper":{"title":"Multi-invariant measures and subsets on nilmanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Zhiren Wang","submitted_at":"2013-09-24T15:48:03Z","abstract_excerpt":"Given a $\\mathbb Z^r$-action $\\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\\alpha$-invariant probability measure $\\mu$, we show that $\\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the following obstructions occurs for an algebraic factor action: 1. The factor measure has zero entropy under every element of the action; 2. The factor action is virtually cyclic. We also deduce a rigidity property for invariant closed subsets."},"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":"1309.6223","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-24T15:48:03Z","cross_cats_sorted":[],"title_canon_sha256":"da1b86f824f3c07a6e09deb50ef5b4f1cba3398c875baf01c9952426ae1366a7","abstract_canon_sha256":"9e3974945f1e9dc40f43702f9aa71d176757f6a83b8c677b422a0f4e98fe85ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:23.454793Z","signature_b64":"GGovkAbVBLfkx1SGqmX0Y113PMkec/Dvg+9IyJXG5/tswu2w1Ueuouy/UYmEM43vFvdxlakuenUYyCKa3SX+Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd9a35135bfb70e79fab9cfe7f0ed1d05c01a5163a0772d0c45963b0f8331b30","last_reissued_at":"2026-05-18T03:12:23.453809Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:23.453809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multi-invariant measures and subsets on nilmanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Zhiren Wang","submitted_at":"2013-09-24T15:48:03Z","abstract_excerpt":"Given a $\\mathbb Z^r$-action $\\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\\alpha$-invariant probability measure $\\mu$, we show that $\\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the following obstructions occurs for an algebraic factor action: 1. The factor measure has zero entropy under every element of the action; 2. The factor action is virtually cyclic. We also deduce a rigidity property for invariant closed subsets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6223","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":"1309.6223","created_at":"2026-05-18T03:12:23.453954+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.6223v1","created_at":"2026-05-18T03:12:23.453954+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6223","created_at":"2026-05-18T03:12:23.453954+00:00"},{"alias_kind":"pith_short_12","alias_value":"7WNDKE237NYO","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"7WNDKE237NYOPH5L","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"7WNDKE23","created_at":"2026-05-18T12:27:38.830355+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/7WNDKE237NYOPH5LTT7H6DWR2B","json":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B.json","graph_json":"https://pith.science/api/pith-number/7WNDKE237NYOPH5LTT7H6DWR2B/graph.json","events_json":"https://pith.science/api/pith-number/7WNDKE237NYOPH5LTT7H6DWR2B/events.json","paper":"https://pith.science/paper/7WNDKE23"},"agent_actions":{"view_html":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B","download_json":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B.json","view_paper":"https://pith.science/paper/7WNDKE23","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.6223&json=true","fetch_graph":"https://pith.science/api/pith-number/7WNDKE237NYOPH5LTT7H6DWR2B/graph.json","fetch_events":"https://pith.science/api/pith-number/7WNDKE237NYOPH5LTT7H6DWR2B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B/action/storage_attestation","attest_author":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B/action/author_attestation","sign_citation":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B/action/citation_signature","submit_replication":"https://pith.science/pith/7WNDKE237NYOPH5LTT7H6DWR2B/action/replication_record"}},"created_at":"2026-05-18T03:12:23.453954+00:00","updated_at":"2026-05-18T03:12:23.453954+00:00"}