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We prove that each ergodic diffeomorphism $ \\phi(x\\Gamma)=uA(x)\\Gamma $ on the nilmanifold $ G/\\Gamma $, where $ u\\in G $ and $ A:G\\to G $ is a unipotent automorphism satisfying $ A(\\Gamma)=\\Gamma $, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow:\n  (i) Sarnak's conjecture on M\\\"obius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism;\n  (ii) For ergodic affine unipotent diffeomorphisms themselves, the M\\\"obius ortho"},"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":"1609.00699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-09-02T18:56:00Z","cross_cats_sorted":[],"title_canon_sha256":"ca7bbb90409beb7425628a4279516edf4a24140e0d8be8cdd3040a23114d97c4","abstract_canon_sha256":"1f2031436634e424f41e3603674ce4482e19243edb79423e84651158802857c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:23.417924Z","signature_b64":"amTWUeBhcPsGuXd6V2rUG/JReT3458tEwhiUuNjvO+k98SQUoVwqhVhol1XAeUtYPNBNrtqmQ46VkY2aOx4kDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4ae27acd0334dc902a6b2fc6f0ec1b4ff9be3a613842d976d73ca0862b4a776","last_reissued_at":"2026-05-18T01:06:23.417233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:23.417233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Joanna Ku{\\l}aga-Przymus, Krzysztof Fr\\k{a}czek, Livio Flaminio, Mariusz Lema\\'nczyk","submitted_at":"2016-09-02T18:56:00Z","abstract_excerpt":"Let $ G $ be a connected, simply connected nilpotent Lie group and $ \\Gamma < G $ a lattice. We prove that each ergodic diffeomorphism $ \\phi(x\\Gamma)=uA(x)\\Gamma $ on the nilmanifold $ G/\\Gamma $, where $ u\\in G $ and $ A:G\\to G $ is a unipotent automorphism satisfying $ A(\\Gamma)=\\Gamma $, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow:\n  (i) Sarnak's conjecture on M\\\"obius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism;\n  (ii) For ergodic affine unipotent diffeomorphisms themselves, the M\\\"obius ortho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00699","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":"1609.00699","created_at":"2026-05-18T01:06:23.417349+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.00699v1","created_at":"2026-05-18T01:06:23.417349+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00699","created_at":"2026-05-18T01:06:23.417349+00:00"},{"alias_kind":"pith_short_12","alias_value":"YSXCPLGQGNG4","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"YSXCPLGQGNG4SAVG","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"YSXCPLGQ","created_at":"2026-05-18T12:30:53.716459+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/YSXCPLGQGNG4SAVGWL6G6DWBWT","json":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT.json","graph_json":"https://pith.science/api/pith-number/YSXCPLGQGNG4SAVGWL6G6DWBWT/graph.json","events_json":"https://pith.science/api/pith-number/YSXCPLGQGNG4SAVGWL6G6DWBWT/events.json","paper":"https://pith.science/paper/YSXCPLGQ"},"agent_actions":{"view_html":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT","download_json":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT.json","view_paper":"https://pith.science/paper/YSXCPLGQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.00699&json=true","fetch_graph":"https://pith.science/api/pith-number/YSXCPLGQGNG4SAVGWL6G6DWBWT/graph.json","fetch_events":"https://pith.science/api/pith-number/YSXCPLGQGNG4SAVGWL6G6DWBWT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT/action/storage_attestation","attest_author":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT/action/author_attestation","sign_citation":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT/action/citation_signature","submit_replication":"https://pith.science/pith/YSXCPLGQGNG4SAVGWL6G6DWBWT/action/replication_record"}},"created_at":"2026-05-18T01:06:23.417349+00:00","updated_at":"2026-05-18T01:06:23.417349+00:00"}