{"paper":{"title":"An inverse of Furstenberg's correspondence principle and applications to nice recurrence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexander Fish, Sean Skinner","submitted_at":"2024-07-28T09:39:22Z","abstract_excerpt":"We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,\\mu,T)$ and $A\\subset X$ measurable there exists a set $E \\subset \\mathbb{N}$ such that \\[ \\mu\\left( \\bigcap_{i=1}^k T^{-n_i}A\\right) = \\lim_{N\\to \\infty} \\frac{\\left|\\left( \\bigcap_{i=1}^k (E-n_i) \\right)\\cap \\{0,\\dots,N-1\\}\\right|}{N}\\] for all $k,n_1,\\dots,n_k \\in \\mathbb{N}$. As a corollary we show that a set $R\\subset \\mathbb{N}$ is a set of nice recurrence if and only if it is nicely intersective. Together, the inverse of Furstenberg's correspondence principle and it's coroll"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.19444","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.19444/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}