{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:IJ3O2ZFBK2KN5NWAMHYWHO4QXY","short_pith_number":"pith:IJ3O2ZFB","canonical_record":{"source":{"id":"1608.02697","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-08-09T05:54:21Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"2e7b8c521c2361d5991387656bc8eed04481b410275524cc64200066db234e37","abstract_canon_sha256":"ea421080e9692371ab091f1ae4099da330c1b97cdeae3c25304a379dd083f1ef"},"schema_version":"1.0"},"canonical_sha256":"4276ed64a15694deb6c061f163bb90be3da47396e24ac3f957eeb075b52b0908","source":{"kind":"arxiv","id":"1608.02697","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.02697","created_at":"2026-05-18T00:33:53Z"},{"alias_kind":"arxiv_version","alias_value":"1608.02697v2","created_at":"2026-05-18T00:33:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02697","created_at":"2026-05-18T00:33:53Z"},{"alias_kind":"pith_short_12","alias_value":"IJ3O2ZFBK2KN","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IJ3O2ZFBK2KN5NWA","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IJ3O2ZFB","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:IJ3O2ZFBK2KN5NWAMHYWHO4QXY","target":"record","payload":{"canonical_record":{"source":{"id":"1608.02697","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-08-09T05:54:21Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"2e7b8c521c2361d5991387656bc8eed04481b410275524cc64200066db234e37","abstract_canon_sha256":"ea421080e9692371ab091f1ae4099da330c1b97cdeae3c25304a379dd083f1ef"},"schema_version":"1.0"},"canonical_sha256":"4276ed64a15694deb6c061f163bb90be3da47396e24ac3f957eeb075b52b0908","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:53.183108Z","signature_b64":"bPxht3UrGvQvU3QHzn66kOvX7V7AquB7kQXVRkgMb2S1OZBqrtsKk3D5oLCfq3KvhhNe7DwcS4+P7LJVS+hKDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4276ed64a15694deb6c061f163bb90be3da47396e24ac3f957eeb075b52b0908","last_reissued_at":"2026-05-18T00:33:53.182572Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:53.182572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.02697","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:33:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rnBtepaKg2r04m4YImXax9zf7uEBwMb14mXqVD6Iy4bvnVSTQP49/mJ8wfMaowPHsaqBuQfVrxxkfkn3+h+IDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:49:27.735048Z"},"content_sha256":"3c61926bca69fb26ec4be2879d65638ca66af5a054fa05480d25fba8ee147ae0","schema_version":"1.0","event_id":"sha256:3c61926bca69fb26ec4be2879d65638ca66af5a054fa05480d25fba8ee147ae0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:IJ3O2ZFBK2KN5NWAMHYWHO4QXY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"M\\\"obius disjointness for non-uniquely ergodic skew products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Zhiren Wang","submitted_at":"2016-08-09T05:54:21Z","abstract_excerpt":"For $\\tau>2$, let $T$ be a $C^\\tau$ skew product map of the form $(x+\\alpha,y+h(x))$ on $\\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the M\\\"{o}bius sequence. This in particular implies that any non-uniquely ergodic $C^\\tau$ skew product map on $\\mathbb T^2$ has a finite index factor that is disjoint to the M\\\"{o}bius sequence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02697","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:33:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5mpILTrrivxDUc/1pOmlfJZHn98P79v9h4pH+HdE1m+suKJYcQYW9TY+NP9v5VZM0BcsKhsGgKscrk9PAhk5Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:49:27.735424Z"},"content_sha256":"7f390becb1033f4810783ba3d5335ff195a4edc31e4b5631126f1af382a87457","schema_version":"1.0","event_id":"sha256:7f390becb1033f4810783ba3d5335ff195a4edc31e4b5631126f1af382a87457"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY/bundle.json","state_url":"https://pith.science/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T06:49:27Z","links":{"resolver":"https://pith.science/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY","bundle":"https://pith.science/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY/bundle.json","state":"https://pith.science/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJ3O2ZFBK2KN5NWAMHYWHO4QXY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IJ3O2ZFBK2KN5NWAMHYWHO4QXY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ea421080e9692371ab091f1ae4099da330c1b97cdeae3c25304a379dd083f1ef","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-08-09T05:54:21Z","title_canon_sha256":"2e7b8c521c2361d5991387656bc8eed04481b410275524cc64200066db234e37"},"schema_version":"1.0","source":{"id":"1608.02697","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.02697","created_at":"2026-05-18T00:33:53Z"},{"alias_kind":"arxiv_version","alias_value":"1608.02697v2","created_at":"2026-05-18T00:33:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02697","created_at":"2026-05-18T00:33:53Z"},{"alias_kind":"pith_short_12","alias_value":"IJ3O2ZFBK2KN","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IJ3O2ZFBK2KN5NWA","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IJ3O2ZFB","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:7f390becb1033f4810783ba3d5335ff195a4edc31e4b5631126f1af382a87457","target":"graph","created_at":"2026-05-18T00:33:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For $\\tau>2$, let $T$ be a $C^\\tau$ skew product map of the form $(x+\\alpha,y+h(x))$ on $\\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the M\\\"{o}bius sequence. This in particular implies that any non-uniquely ergodic $C^\\tau$ skew product map on $\\mathbb T^2$ has a finite index factor that is disjoint to the M\\\"{o}bius sequence.","authors_text":"Zhiren Wang","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-08-09T05:54:21Z","title":"M\\\"obius disjointness for non-uniquely ergodic skew products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02697","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3c61926bca69fb26ec4be2879d65638ca66af5a054fa05480d25fba8ee147ae0","target":"record","created_at":"2026-05-18T00:33:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ea421080e9692371ab091f1ae4099da330c1b97cdeae3c25304a379dd083f1ef","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-08-09T05:54:21Z","title_canon_sha256":"2e7b8c521c2361d5991387656bc8eed04481b410275524cc64200066db234e37"},"schema_version":"1.0","source":{"id":"1608.02697","kind":"arxiv","version":2}},"canonical_sha256":"4276ed64a15694deb6c061f163bb90be3da47396e24ac3f957eeb075b52b0908","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4276ed64a15694deb6c061f163bb90be3da47396e24ac3f957eeb075b52b0908","first_computed_at":"2026-05-18T00:33:53.182572Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:53.182572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bPxht3UrGvQvU3QHzn66kOvX7V7AquB7kQXVRkgMb2S1OZBqrtsKk3D5oLCfq3KvhhNe7DwcS4+P7LJVS+hKDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:53.183108Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.02697","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c61926bca69fb26ec4be2879d65638ca66af5a054fa05480d25fba8ee147ae0","sha256:7f390becb1033f4810783ba3d5335ff195a4edc31e4b5631126f1af382a87457"],"state_sha256":"64a7e0894c3be7d6e56f158c7a545d517d35e12f06675270082eb9d6a4d60dd1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N4Yt3mWziwizv/Q+dmyoagByF6jfhIeHTbF5/eoqsSELTi05RKPplweIf5LxuZM72Od/+GeKIhSnX0xkFiynAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T06:49:27.737324Z","bundle_sha256":"ea4a6253ac3fb3119882c5a7fe5f8090aeee49322524ebe0b081de7f2acd5dff"}}