{
  "id": "1507.01276",
  "version": "v1",
  "published": "2015-07-05T21:33:27.000Z",
  "updated": "2015-07-05T21:33:27.000Z",
  "title": "Inverse theorems for sets and measures of polynomial growth",
  "authors": [
    "Terence Tao"
  ],
  "comment": "42 pages, no figures",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \\leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of $|A^m|$ fairly explicitly for $m \\geq n$, at least when $A$ is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures $\\mu$ whose $n$-fold convolutions $\\mu^{*n}$ obey the condition $\\| \\mu^{*n} \\|_{\\ell^2}^{-2} \\leq n^d \\|\\mu \\|_{\\ell^2}^{-2}$. In the abelian case, this description recovers the inverse Littlewood-Offord theorem of Nguyen-Vu, and gives a variant of a recent nonabelian inverse Littlewood-Offord theorem of Tiep-Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-05T21:33:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "60G50"
    ],
    "keywords": [
      "inverse theorem",
      "nonabelian inverse littlewood-offord theorem",
      "polynomial growth condition",
      "symmetric probability measures",
      "main tool"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150701276T"
    }
  }
}