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As a consequence every mixing time change is mixing of all orders. Moreover we show that for every $\\tau\\in W^s(M)$, $s>9/2$ and every $p,q\\in \\mathbb{N}$, $p\\neq q$, $(\\phi_{pt}^\\tau)$ and $(\\phi_{qt}^\\tau)$ are disjoint. As a consequence Sarnak's Conjecture on M\\\"obius disjointness holds for all such time changes."},"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":"1810.13319","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-10-31T14:55:17Z","cross_cats_sorted":[],"title_canon_sha256":"e1f563ac28ed034edd93e57410f65ddb84b4d71a3b96fe5ea0e433907ea4c5c8","abstract_canon_sha256":"f8dfc90681e28e8f74b27ff1021ad2738c2040ba2226df204da5884b71f71cd5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:49.716007Z","signature_b64":"IQM21IqSHMD0hsII4Q9K6N2AvyO98y/5MqNInUGRiYfPBP9VcDCMMAuh58C/t5p/NiY25ax+9JBJp0IvBv2ODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ed8173043bf93bf728ac40b1834ca0ebe3f8c99536e851dd9269cac08406122","last_reissued_at":"2026-05-18T00:01:49.715536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:49.715536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mutliple mixing and disjointness for time changes of bounded-type Heisenberg nilflows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Adam Kanigowski, Giovanni Forni","submitted_at":"2018-10-31T14:55:17Z","abstract_excerpt":"We study time changes of bounded type Heisenberg nilflows $(\\phi_t)$ acting on the Heisenberg nilmanifold $M$. We show that for every positive $\\tau\\in W^s(M)$, $s~>~7/2$, every non-trivial time change $(\\phi_t^{\\tau})$ enjoys the Ratner property. As a consequence every mixing time change is mixing of all orders. Moreover we show that for every $\\tau\\in W^s(M)$, $s>9/2$ and every $p,q\\in \\mathbb{N}$, $p\\neq q$, $(\\phi_{pt}^\\tau)$ and $(\\phi_{qt}^\\tau)$ are disjoint. 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