{"paper":{"title":"Under recurrence in the Khintchine recurrence theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"M\\'at\\'e Wierdl, Michael Boshernitzan, Nikos Frantzikinakis","submitted_at":"2016-03-24T19:49:17Z","abstract_excerpt":"The Khintchine recurrence theorem asserts that on a measure preserving system, for every set $A$ and $\\varepsilon>0$, we have $\\mu(A\\cap T^{-n}A)\\geq \\mu(A)^2-\\varepsilon$ for infinitely many $n\\in \\mathbb{N}$. We show that there are systems having under-recurrent sets $A$, in the sense that the inequality $\\mu(A\\cap T^{-n}A)< \\mu(A)^2$ holds for every $n\\in \\mathbb{N}$. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V.~Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07720","kind":"arxiv","version":3},"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"}