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Here is a sample of results obtained in this paper:\n  $\\cdot$ (Existence of density-1 UI and OI set) Let $(X,\\mathcal{B},\\mu,T)$ be an invertible probability measure preserving weakly mixing system. Then for any $d\\in\\mathbb{N}$, any non-constant integer-valued polynomials $p_{1},p_{2},\\dots,p_{d}$ such that $p_{i}-p_{j}$ are also non-constant for all $i\\neq j$,\n  (i) there is $A\\i"},"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":"1807.02966","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-07-09T07:02:34Z","cross_cats_sorted":[],"title_canon_sha256":"cc6731d9b855241bf0f18c531d2e4b31a1a1c7a123038bf47c6a0d6e7187d47c","abstract_canon_sha256":"db8a5910313ea37ac4b1ae364b775d9a3e64bd477791022e08e446343c39b4c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:59.206798Z","signature_b64":"tH8LmFL5VvdUxhiX0Tzkxqn8mt1yy6h9sB3bhLLfwdcaLkvRzcweA6pVJ4ONHpugKUij+RI+vZ9o0TVyJbT9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76f48847364746e62554e7776da95c5ceaf0510cd81f7112f92be8b5023c2317","last_reissued_at":"2026-05-18T00:10:59.206074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:59.206074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Under- and over-independence in measure preserving systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Terry Adams, Vitaly Bergelson, Wenbo Sun","submitted_at":"2018-07-09T07:02:34Z","abstract_excerpt":"We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. 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