{"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]. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02966","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"}