{
  "id": "1701.08139",
  "version": "v1",
  "published": "2017-01-27T18:26:28.000Z",
  "updated": "2017-01-27T18:26:28.000Z",
  "title": "Quantitative multiple recurrence for two and three transformations",
  "authors": [
    "Sebastián Donoso",
    "Wenbo Sun"
  ],
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $\\bullet$ 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 $$\\mu(A\\cap T_{1}^{-n}A\\cap T_{2}^{-n}A)<\\mu(A)^{\\ell} \\text{ for every } n\\neq 0;$$ $\\bullet$ 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 $$\\mu(A\\cap T_{1}^{-n}A\\cap T_{2}^{-n}A\\cap T_{3}^{-n}A)<\\mu(A)^{\\ell} \\text{ for every } n\\neq 0;$$ $\\bullet$ There exists an ergodic system $(X,\\mathcal{X},\\mu,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $\\ell>0$, there exists $A\\in\\mathcal{X}$ such that $$\\mu(A\\cap T_{1}^{-n}A\\cap T_{2}^{-n}A)<\\mu(A)^{\\ell} \\text{ for every } n\\neq 0.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-27T18:26:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "05A20"
    ],
    "keywords": [
      "ergodic system",
      "quantitative multiple recurrence problems",
      "commuting transformations",
      "counter examples",
      "nilpotent group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}