{"paper":{"title":"Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Valery V. Ryzhikov","submitted_at":"2024-11-20T17:35:26Z","abstract_excerpt":"We recall theorems by Krygin, Atkinson, Shneiberg and propose the following assertion. Let $T_t$ be an ergodic flow on $(X,\\mu)$, let a function $f$ on $X$ have zero mean, and $\\mu(A)>0$ for $A\\subset X$. Then for almost all $x\\in A$ with $f(x)\\neq 0$ there exists a sequence $t_k\\to\\infty$ such that $\\int_0^{t_k}f(T_sx)ds=0$ and $T_{t_k}x\\in A$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13486","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.13486/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"}