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Then for any $\\epsilon>0$ and $A\\subseteq X$ with $\\mu(A)>0$, the set $$ \\{n:\\,\\mu(A\\cap T_1^{-p_1(n)}A\\cap...\\cap T_l^{-p_l(n)}A)\\geq\\mu(A)^{l+1}-\\epsilon\\} $$ has bounded gaps."},"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":"1502.07203","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-25T15:37:19Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"cc3c032ee43ae099d38072be811eb9e072aba3ae8f1fa34d824a29890790f4a2","abstract_canon_sha256":"c375e4ee14644745f3687165a6a99197f633d72caad147df431ad048e6f3209e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:10.028027Z","signature_b64":"jcPhCACuc3xtGgqlvvOHMtc52giSZT8PEK4I0G/UBp+4F5mawvIEE73HF8+fNL0pbCvxKl09lBKlXwFAfrzJCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b68c36ff9061217a01d01a7b6f28db176537e01ac404d87dcdd5e4d9603859b","last_reissued_at":"2026-05-18T02:26:10.027479Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:10.027479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Note on polynomial recurrence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CO","authors_text":"Hao Pan","submitted_at":"2015-02-25T15:37:19Z","abstract_excerpt":"Let $(X,\\mu,T_1,...,T_l)$ be a measure-preserving system with those $T_i$ are commuting. 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