{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:PXAJXMWTL4SP7H7GE3HXEU5SNQ","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":"f04570c38625ba8ae1e3d9b96b35740419333824b4e6a3a1837b3eefd3ccaa52","cross_cats_sorted":["math.CO","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-09-16T14:49:50Z","title_canon_sha256":"a22c94e6dd68c1a370aeaf2215f1dcf498419bd9a7a9e6fa8bc28d8e5a18e77d"},"schema_version":"1.0","source":{"id":"1109.3636","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.3636","created_at":"2026-05-18T04:12:55Z"},{"alias_kind":"arxiv_version","alias_value":"1109.3636v1","created_at":"2026-05-18T04:12:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.3636","created_at":"2026-05-18T04:12:55Z"},{"alias_kind":"pith_short_12","alias_value":"PXAJXMWTL4SP","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PXAJXMWTL4SP7H7G","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PXAJXMWT","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:1b1f2af769c368fc76df37b3be23a3f87352e50f542653e979e47591bac8e1a5","target":"graph","created_at":"2026-05-18T04:12:55Z","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":"The problem which can be viewed as the higher order version of an old question concerning Bohr sets is investigated: for any $d\\in \\N$ does the collection of $\\{n\\in \\Z: S\\cap (S-n)\\cap...\\cap (S-dn)\\neq \\emptyset\\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets?\n  In this paper it is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem could be answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\\supset \\{n\\in \\Z: S\\cap (S-n)\\cap...\\cap (S-dn)\\neq","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","cross_cats":["math.CO","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-09-16T14:49:50Z","title":"Nil Bohr$_0$-sets, Poincar\\'e recurrence and generalized polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3636","kind":"arxiv","version":1},"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:aacc09aebfa9c035dc42d8376f18fc408da84f0bf2aed9c667e2ba50745bfeeb","target":"record","created_at":"2026-05-18T04:12:55Z","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":"f04570c38625ba8ae1e3d9b96b35740419333824b4e6a3a1837b3eefd3ccaa52","cross_cats_sorted":["math.CO","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-09-16T14:49:50Z","title_canon_sha256":"a22c94e6dd68c1a370aeaf2215f1dcf498419bd9a7a9e6fa8bc28d8e5a18e77d"},"schema_version":"1.0","source":{"id":"1109.3636","kind":"arxiv","version":1}},"canonical_sha256":"7dc09bb2d35f24ff9fe626cf7253b26c2ec2acd17894a49d06b8a668e0edc9bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7dc09bb2d35f24ff9fe626cf7253b26c2ec2acd17894a49d06b8a668e0edc9bc","first_computed_at":"2026-05-18T04:12:55.434377Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:55.434377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3L0NfpItDewK6hf6RfE/D3+yAjltwsBrsPtYQQ219sPQt7eHzyEBEPQx+WI+izMR3ecc9VMa1+2Q3XfzFNNTAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:55.434933Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.3636","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aacc09aebfa9c035dc42d8376f18fc408da84f0bf2aed9c667e2ba50745bfeeb","sha256:1b1f2af769c368fc76df37b3be23a3f87352e50f542653e979e47591bac8e1a5"],"state_sha256":"48fd27017fb8f5725ce3ec4240532ae46e3e2847bb5223d1405f0fe238970999"}