{
  "id": "1405.3132",
  "version": "v1",
  "published": "2014-05-13T12:53:41.000Z",
  "updated": "2014-05-13T12:53:41.000Z",
  "title": "Energies and structure of additive sets",
  "authors": [
    "Ilya D. Shkredov"
  ],
  "comment": "52 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we give criteria for a set to be a 1) set of the form H+L, where H+H is small and L has \"random structure\", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \\approx |A|^4 / \\E(A).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-13T12:53:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "additive sets",
      "disjoint union",
      "difference set",
      "large subset",
      "set equals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3132S"
    }
  }
}