{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YBBJOOYNWQIK5KSJUU5NNOLOLE","short_pith_number":"pith:YBBJOOYN","canonical_record":{"source":{"id":"1203.3778","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-16T18:36:33Z","cross_cats_sorted":[],"title_canon_sha256":"7be63384cac045bb6b92707d432e2d8e4c35032834969f24684789a80c420185","abstract_canon_sha256":"bdbee4f9298c9f693444375d592c71b18a3dbae0f8c37dcaf134ed4e87113485"},"schema_version":"1.0"},"canonical_sha256":"c042973b0db410aeaa49a53ad6b96e5914779ac26a33611de2625d848491e665","source":{"kind":"arxiv","id":"1203.3778","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.3778","created_at":"2026-05-18T03:55:46Z"},{"alias_kind":"arxiv_version","alias_value":"1203.3778v2","created_at":"2026-05-18T03:55:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.3778","created_at":"2026-05-18T03:55:46Z"},{"alias_kind":"pith_short_12","alias_value":"YBBJOOYNWQIK","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YBBJOOYNWQIK5KSJ","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YBBJOOYN","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YBBJOOYNWQIK5KSJUU5NNOLOLE","target":"record","payload":{"canonical_record":{"source":{"id":"1203.3778","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-16T18:36:33Z","cross_cats_sorted":[],"title_canon_sha256":"7be63384cac045bb6b92707d432e2d8e4c35032834969f24684789a80c420185","abstract_canon_sha256":"bdbee4f9298c9f693444375d592c71b18a3dbae0f8c37dcaf134ed4e87113485"},"schema_version":"1.0"},"canonical_sha256":"c042973b0db410aeaa49a53ad6b96e5914779ac26a33611de2625d848491e665","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:46.380712Z","signature_b64":"4xGr2u3Fly3s4llL2qtE/prhCLTAnBbHwzTP40zHqfjLf2vHHmIe2kWSCpvhRivbYyw7tHz+oGcdggPeaHzFCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c042973b0db410aeaa49a53ad6b96e5914779ac26a33611de2625d848491e665","last_reissued_at":"2026-05-18T03:55:46.380296Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:46.380296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.3778","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":"2026-05-18T03:55:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oRjKIXxEN41h7xL5G4L3Tnrv4cdEPMWCGN+isGp5//b/99y4P4bFaBt1oMGr/5czMK0MbKtRJu5V2Pwr7asyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T13:33:53.555674Z"},"content_sha256":"19852b93d84bcb699af24fcd57accbff680f52e061af1621ed59d6c1936a9175","schema_version":"1.0","event_id":"sha256:19852b93d84bcb699af24fcd57accbff680f52e061af1621ed59d6c1936a9175"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YBBJOOYNWQIK5KSJUU5NNOLOLE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Complexity of Nilsystems and systems lacking nilfactors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alejandro Maass, Bernard Host, Bryna Kra","submitted_at":"2012-03-16T18:36:33Z","abstract_excerpt":"Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3778","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":"2026-05-18T03:55:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5dcVBJsjqadtYDi/1MFd+Vm6fdmXax4dJFA+8GappkRhJH0xJoX5JlAPO8jA/WGT++d9i1zTU97bDJ7RUooWCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T13:33:53.556016Z"},"content_sha256":"99e0941faea574f2de14e5628699eb91e4cd0b1e8ec3ad73a041df66e9d8ecda","schema_version":"1.0","event_id":"sha256:99e0941faea574f2de14e5628699eb91e4cd0b1e8ec3ad73a041df66e9d8ecda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE/bundle.json","state_url":"https://pith.science/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE/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":"2026-06-22T13:33:53Z","links":{"resolver":"https://pith.science/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE","bundle":"https://pith.science/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE/bundle.json","state":"https://pith.science/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YBBJOOYNWQIK5KSJUU5NNOLOLE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YBBJOOYNWQIK5KSJUU5NNOLOLE","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":"bdbee4f9298c9f693444375d592c71b18a3dbae0f8c37dcaf134ed4e87113485","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-16T18:36:33Z","title_canon_sha256":"7be63384cac045bb6b92707d432e2d8e4c35032834969f24684789a80c420185"},"schema_version":"1.0","source":{"id":"1203.3778","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.3778","created_at":"2026-05-18T03:55:46Z"},{"alias_kind":"arxiv_version","alias_value":"1203.3778v2","created_at":"2026-05-18T03:55:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.3778","created_at":"2026-05-18T03:55:46Z"},{"alias_kind":"pith_short_12","alias_value":"YBBJOOYNWQIK","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YBBJOOYNWQIK5KSJ","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YBBJOOYN","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:99e0941faea574f2de14e5628699eb91e4cd0b1e8ec3ad73a041df66e9d8ecda","target":"graph","created_at":"2026-05-18T03:55:46Z","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":"Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise ","authors_text":"Alejandro Maass, Bernard Host, Bryna Kra","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-16T18:36:33Z","title":"Complexity of Nilsystems and systems lacking nilfactors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.3778","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":"sha256:19852b93d84bcb699af24fcd57accbff680f52e061af1621ed59d6c1936a9175","target":"record","created_at":"2026-05-18T03:55:46Z","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":"bdbee4f9298c9f693444375d592c71b18a3dbae0f8c37dcaf134ed4e87113485","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-03-16T18:36:33Z","title_canon_sha256":"7be63384cac045bb6b92707d432e2d8e4c35032834969f24684789a80c420185"},"schema_version":"1.0","source":{"id":"1203.3778","kind":"arxiv","version":2}},"canonical_sha256":"c042973b0db410aeaa49a53ad6b96e5914779ac26a33611de2625d848491e665","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c042973b0db410aeaa49a53ad6b96e5914779ac26a33611de2625d848491e665","first_computed_at":"2026-05-18T03:55:46.380296Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:46.380296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4xGr2u3Fly3s4llL2qtE/prhCLTAnBbHwzTP40zHqfjLf2vHHmIe2kWSCpvhRivbYyw7tHz+oGcdggPeaHzFCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:46.380712Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.3778","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19852b93d84bcb699af24fcd57accbff680f52e061af1621ed59d6c1936a9175","sha256:99e0941faea574f2de14e5628699eb91e4cd0b1e8ec3ad73a041df66e9d8ecda"],"state_sha256":"14711cb68da071f66be1b3d24f10fcb3a0c76657b6a65b520e5a8055468ed780"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nq3ZgNcSX7AsetlAPwNpdT7H/pUfOwFLrhT2rPx+VFFz3TUsJPCrEGUB+0UbgQfWjvOG9+ztWrp3oE1Am1/TBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T13:33:53.558025Z","bundle_sha256":"61d122474f0b0b053ae879b136ef9884c7cf3b9d9816c058de3a6ce33c7a3645"}}