{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:CTCICHMZCZSAZBER7PTQJWCW7M","short_pith_number":"pith:CTCICHMZ","schema_version":"1.0","canonical_sha256":"14c4811d9916640c8491fbe704d856fb32ff0d830f6b483df51307d37bdd88fe","source":{"kind":"arxiv","id":"2407.10728","version":1},"attestation_state":"computed","paper":{"title":"A counterexample on multiple convergence without commutativity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","submitted_at":"2024-07-15T13:53:17Z","abstract_excerpt":"It is shown that there exist a probability space $(X,{\\mathcal X},\\mu)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\\mathcal X},\\mu)$ with $h_\\mu(X,T)=h_\\mu(X,S)=0$, and $f, g \\in L^\\infty(X,\\mu)$ such that the limit \\begin{equation*}\n  \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \\end{equation*} does not exist in $L^2(X,\\mu)$."},"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":"2407.10728","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2024-07-15T13:53:17Z","cross_cats_sorted":[],"title_canon_sha256":"cdc1e16c6778c1d13b7adfb63fca2db8e8f720668cf4f44233c0b57b76a7c759","abstract_canon_sha256":"3e14c24facce9bd7af9ddbbcbd04f18d0c29d040122a295ac3678f802bc91353"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:44:03.370270Z","signature_b64":"If1Sk5IVBJRO0vJHhsJHQahtMYH/nF3n+xqw6wRwZe+/+AklLydYoyCesnAbQ8GItk4EoyOopR5g/5hREDU/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14c4811d9916640c8491fbe704d856fb32ff0d830f6b483df51307d37bdd88fe","last_reissued_at":"2026-07-05T08:44:03.369802Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:44:03.369802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A counterexample on multiple convergence without commutativity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","submitted_at":"2024-07-15T13:53:17Z","abstract_excerpt":"It is shown that there exist a probability space $(X,{\\mathcal X},\\mu)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\\mathcal X},\\mu)$ with $h_\\mu(X,T)=h_\\mu(X,S)=0$, and $f, g \\in L^\\infty(X,\\mu)$ such that the limit \\begin{equation*}\n  \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \\end{equation*} does not exist in $L^2(X,\\mu)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.10728","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.10728/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2407.10728","created_at":"2026-07-05T08:44:03.369859+00:00"},{"alias_kind":"arxiv_version","alias_value":"2407.10728v1","created_at":"2026-07-05T08:44:03.369859+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.10728","created_at":"2026-07-05T08:44:03.369859+00:00"},{"alias_kind":"pith_short_12","alias_value":"CTCICHMZCZSA","created_at":"2026-07-05T08:44:03.369859+00:00"},{"alias_kind":"pith_short_16","alias_value":"CTCICHMZCZSAZBER","created_at":"2026-07-05T08:44:03.369859+00:00"},{"alias_kind":"pith_short_8","alias_value":"CTCICHMZ","created_at":"2026-07-05T08:44:03.369859+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.15528","citing_title":"Counterexamples to double recurrence for non-commuting deterministic transformations","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M","json":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M.json","graph_json":"https://pith.science/api/pith-number/CTCICHMZCZSAZBER7PTQJWCW7M/graph.json","events_json":"https://pith.science/api/pith-number/CTCICHMZCZSAZBER7PTQJWCW7M/events.json","paper":"https://pith.science/paper/CTCICHMZ"},"agent_actions":{"view_html":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M","download_json":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M.json","view_paper":"https://pith.science/paper/CTCICHMZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2407.10728&json=true","fetch_graph":"https://pith.science/api/pith-number/CTCICHMZCZSAZBER7PTQJWCW7M/graph.json","fetch_events":"https://pith.science/api/pith-number/CTCICHMZCZSAZBER7PTQJWCW7M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M/action/storage_attestation","attest_author":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M/action/author_attestation","sign_citation":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M/action/citation_signature","submit_replication":"https://pith.science/pith/CTCICHMZCZSAZBER7PTQJWCW7M/action/replication_record"}},"created_at":"2026-07-05T08:44:03.369859+00:00","updated_at":"2026-07-05T08:44:03.369859+00:00"}