{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:RQCPLFCUWAJNMULQXWQJMPHNL7","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":"8be904c984a462423be80780c5da70d05324be7e37e720de57c0fa59143e7976","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-22T21:18:47Z","title_canon_sha256":"e592a8f1dc261d3ad31fe48d4b6ff9990455be43388bac5e8dbbeb52b1761d1b"},"schema_version":"1.0","source":{"id":"1001.4081","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.4081","created_at":"2026-05-18T01:56:00Z"},{"alias_kind":"arxiv_version","alias_value":"1001.4081v2","created_at":"2026-05-18T01:56:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.4081","created_at":"2026-05-18T01:56:00Z"},{"alias_kind":"pith_short_12","alias_value":"RQCPLFCUWAJN","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RQCPLFCUWAJNMULQ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RQCPLFCU","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:779a866953ee9257f6782cb3c15ed8b84c9c67b0f11c13ab506fc90f21ba5671","target":"graph","created_at":"2026-05-18T01:56:00Z","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":"Let $E\\subset \\mathbb Z$ be a set of positive upper density. Suppose that $P_1,P_2,..., P_k\\in \\mathbb Z[X]$ are polynomials having zero constant terms. We show that the set $E\\cap (E-P_1(p-1))\\cap ... \\cap (E-P_k(p-1))$ is non-empty for some prime number $p$. Furthermore, we prove convergence in $L^2$ of polynomial multiple averages along the primes.","authors_text":"Tamar D. Ziegler, Trevor D. Wooley","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-22T21:18:47Z","title":"Multiple recurrence and convergence along the primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4081","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":"sha256:7c73f4c26383b4fa0ac15de6ec81afef0c39f069434b1b8859331efd31968242","target":"record","created_at":"2026-05-18T01:56:00Z","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":"8be904c984a462423be80780c5da70d05324be7e37e720de57c0fa59143e7976","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-22T21:18:47Z","title_canon_sha256":"e592a8f1dc261d3ad31fe48d4b6ff9990455be43388bac5e8dbbeb52b1761d1b"},"schema_version":"1.0","source":{"id":"1001.4081","kind":"arxiv","version":2}},"canonical_sha256":"8c04f59454b012d65170bda0963ced5fc863f9ed65cfb980a18d879ea3ea7b7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c04f59454b012d65170bda0963ced5fc863f9ed65cfb980a18d879ea3ea7b7a","first_computed_at":"2026-05-18T01:56:00.235022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:56:00.235022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eju5JgCfONYTLYBP7meSTO4jkoUkHXxtFik9ZTcJqNLAJbpignOusSfNox/5jpl1ZmI9m/ygJ1bZILAZ7FJeDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:56:00.235601Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.4081","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c73f4c26383b4fa0ac15de6ec81afef0c39f069434b1b8859331efd31968242","sha256:779a866953ee9257f6782cb3c15ed8b84c9c67b0f11c13ab506fc90f21ba5671"],"state_sha256":"ce5881f4fe8637230008464427002be041960081065a78e432b055902d29244b"}