{
  "id": "1907.12686",
  "version": "v1",
  "published": "2019-07-29T23:23:40.000Z",
  "updated": "2019-07-29T23:23:40.000Z",
  "title": "Concentration of measure, classification of submeasures, and dynamics of $L_{0}$",
  "authors": [
    "Friedrich Martin Schneider",
    "Sławomir Solecki"
  ],
  "categories": [
    "math.PR",
    "math.DS",
    "math.LO"
  ],
  "abstract": "Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an entropic version of the weighted Loomis--Whitney inequality. We give a quantitative \"geometric\" classification of diffused submeasures into elliptic, parabolic, and hyperbolic. We prove that any non-elliptic submeasure (for example, any measure, or any pathological submeasure) has a property that we call covering concentration. Our results have strong consequences for the dynamics of the corresponding topological $L_0$-groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-29T23:23:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "28A60",
      "43A07",
      "54H15"
    ],
    "keywords": [
      "classification",
      "uniform concentration bounds",
      "herbst argument",
      "strong consequences",
      "non-elliptic submeasure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}