{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:P75V6NQMZ3GJX5WSPCT3CM7CKS","short_pith_number":"pith:P75V6NQM","schema_version":"1.0","canonical_sha256":"7ffb5f360ccecc9bf6d278a7b133e2548cc0ea6e817f8aa862059f5ae44aa39e","source":{"kind":"arxiv","id":"1307.5471","version":3},"attestation_state":"computed","paper":{"title":"Mean Dimension, Mean Rank, and von Neumann-L\\\"uck Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.FA","math.GR","math.OA"],"primary_cat":"math.DS","authors_text":"Bingbing Liang, Hanfeng Li","submitted_at":"2013-07-20T22:34:04Z","abstract_excerpt":"We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $\\Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $\\Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-L\\\"uck rank of M all coincide.\n  As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of fi"},"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":"1307.5471","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-07-20T22:34:04Z","cross_cats_sorted":["math.AT","math.FA","math.GR","math.OA"],"title_canon_sha256":"d65dc904e3c3fcff2819a330044c6ea6139505588e16ae9a037ce073f62d14c4","abstract_canon_sha256":"b147185e4b90a2ed6a8ca0e786ddfab0bffc0f97719081e42256ffbb145e8dee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:01.327369Z","signature_b64":"SM2EpjLAgIhJ2hhQrZcfEiz8FIwc4aUpny0pbPoWkSamqSrH0Gwz54gsT2ME2fDwC02G4O6kktfCf3Kapr1qBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ffb5f360ccecc9bf6d278a7b133e2548cc0ea6e817f8aa862059f5ae44aa39e","last_reissued_at":"2026-05-18T00:12:01.326712Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:01.326712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mean Dimension, Mean Rank, and von Neumann-L\\\"uck Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.FA","math.GR","math.OA"],"primary_cat":"math.DS","authors_text":"Bingbing Liang, Hanfeng Li","submitted_at":"2013-07-20T22:34:04Z","abstract_excerpt":"We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $\\Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $\\Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-L\\\"uck rank of M all coincide.\n  As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5471","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1307.5471","created_at":"2026-05-18T00:12:01.326821+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.5471v3","created_at":"2026-05-18T00:12:01.326821+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5471","created_at":"2026-05-18T00:12:01.326821+00:00"},{"alias_kind":"pith_short_12","alias_value":"P75V6NQMZ3GJ","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"P75V6NQMZ3GJX5WS","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"P75V6NQM","created_at":"2026-05-18T12:27:54.935989+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/P75V6NQMZ3GJX5WSPCT3CM7CKS","json":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS.json","graph_json":"https://pith.science/api/pith-number/P75V6NQMZ3GJX5WSPCT3CM7CKS/graph.json","events_json":"https://pith.science/api/pith-number/P75V6NQMZ3GJX5WSPCT3CM7CKS/events.json","paper":"https://pith.science/paper/P75V6NQM"},"agent_actions":{"view_html":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS","download_json":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS.json","view_paper":"https://pith.science/paper/P75V6NQM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.5471&json=true","fetch_graph":"https://pith.science/api/pith-number/P75V6NQMZ3GJX5WSPCT3CM7CKS/graph.json","fetch_events":"https://pith.science/api/pith-number/P75V6NQMZ3GJX5WSPCT3CM7CKS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS/action/storage_attestation","attest_author":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS/action/author_attestation","sign_citation":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS/action/citation_signature","submit_replication":"https://pith.science/pith/P75V6NQMZ3GJX5WSPCT3CM7CKS/action/replication_record"}},"created_at":"2026-05-18T00:12:01.326821+00:00","updated_at":"2026-05-18T00:12:01.326821+00:00"}