{"paper":{"title":"Quantitative multiple recurrence for two and three transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Sebasti\\'an Donoso, Wenbo Sun","submitted_at":"2017-01-27T18:26:28Z","abstract_excerpt":"We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation.\n  We show that\n  $\\bullet$ There exists an ergodic system $(X,\\mathcal{X},\\mu,T_1,T_2)$ with two commuting transformations such that for every $0<\\ell< 4$, there exists $A\\in\\mathcal{X}$ such that $$\\mu(A\\cap T_{1}^{-n}A\\cap T_{2}^{-n}A)<\\mu(A)^{\\ell} \\text{ for every } n\\neq 0;$$\n  $\\bullet$ There exists an ergodic system $(X,\\mathcal{X},\\mu,T_1,T_2, T_{3})$ with three commuting transformations such that for every $\\ell>0$, there exists $A\\in\\mathcal{X}$ such that $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08139","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":""},"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"}