{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:IAF7SLWXSQC25XPTL4OF6C5B7H","short_pith_number":"pith:IAF7SLWX","schema_version":"1.0","canonical_sha256":"400bf92ed79405aeddf35f1c5f0ba1f9e6ea9fa0b1c09cf3a97e09666c8745f7","source":{"kind":"arxiv","id":"1104.1961","version":2},"attestation_state":"computed","paper":{"title":"A survey on spectral multiplicities of ergodic actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexandre I. Danilenko","submitted_at":"2011-04-11T14:38:56Z","abstract_excerpt":"Given a transformation $T$ of a standard measure space $(X,\\mu)$, let $\\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,\\mu)\\ominus\\Bbb C$ by $U_Tf:=f\\circ T$. It is discussed in this survey paper which subsets of $\\Bbb N\\cup\\{\\infty\\}$ are realizable as $\\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed."},"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":"1104.1961","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-04-11T14:38:56Z","cross_cats_sorted":[],"title_canon_sha256":"516518b28302b68a667f34228f86fd711b1b519bd3628e68a11e0c3015c32414","abstract_canon_sha256":"ccbc46715f7b3769011ee004f2928e3770bedbdfd213991215dd285c95979313"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:47.783938Z","signature_b64":"GCby7AEPYDsnEMhM8p28Co4AbFC/tRmNmvRQExUUhvvzjIW6pc6VLzGgC0oR70p51cOby9zNFWf3Cw+OK2JpDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"400bf92ed79405aeddf35f1c5f0ba1f9e6ea9fa0b1c09cf3a97e09666c8745f7","last_reissued_at":"2026-05-18T04:15:47.783342Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:47.783342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A survey on spectral multiplicities of ergodic actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexandre I. Danilenko","submitted_at":"2011-04-11T14:38:56Z","abstract_excerpt":"Given a transformation $T$ of a standard measure space $(X,\\mu)$, let $\\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,\\mu)\\ominus\\Bbb C$ by $U_Tf:=f\\circ T$. It is discussed in this survey paper which subsets of $\\Bbb N\\cup\\{\\infty\\}$ are realizable as $\\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1961","kind":"arxiv","version":2},"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":"1104.1961","created_at":"2026-05-18T04:15:47.783441+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.1961v2","created_at":"2026-05-18T04:15:47.783441+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.1961","created_at":"2026-05-18T04:15:47.783441+00:00"},{"alias_kind":"pith_short_12","alias_value":"IAF7SLWXSQC2","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"IAF7SLWXSQC25XPT","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"IAF7SLWX","created_at":"2026-05-18T12:26:30.835961+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/IAF7SLWXSQC25XPTL4OF6C5B7H","json":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H.json","graph_json":"https://pith.science/api/pith-number/IAF7SLWXSQC25XPTL4OF6C5B7H/graph.json","events_json":"https://pith.science/api/pith-number/IAF7SLWXSQC25XPTL4OF6C5B7H/events.json","paper":"https://pith.science/paper/IAF7SLWX"},"agent_actions":{"view_html":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H","download_json":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H.json","view_paper":"https://pith.science/paper/IAF7SLWX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.1961&json=true","fetch_graph":"https://pith.science/api/pith-number/IAF7SLWXSQC25XPTL4OF6C5B7H/graph.json","fetch_events":"https://pith.science/api/pith-number/IAF7SLWXSQC25XPTL4OF6C5B7H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H/action/storage_attestation","attest_author":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H/action/author_attestation","sign_citation":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H/action/citation_signature","submit_replication":"https://pith.science/pith/IAF7SLWXSQC25XPTL4OF6C5B7H/action/replication_record"}},"created_at":"2026-05-18T04:15:47.783441+00:00","updated_at":"2026-05-18T04:15:47.783441+00:00"}