{
  "id": "1711.04337",
  "version": "v1",
  "published": "2017-11-12T18:43:15.000Z",
  "updated": "2017-11-12T18:43:15.000Z",
  "title": "An inverse theorem for the Kemperman inequality",
  "authors": [
    "Terence Tao"
  ],
  "comment": "24 pages, no figures. Submitted, Proceedings of the Steklov Institute of Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (G,+)$ be a compact connected abelian group, and let $\\mu_G$ denote its probability Haar measure. A theorem of Kemperman (generalising previous results of Macbeath and Raikov) establishes the bound $$ \\mu_G(A + B) \\geq \\min( \\mu_G(A)+\\mu_G(B), 1 ) $$ whenever $A,B$ are compact subsets of $G$, and $A+B := \\{ a+b: a \\in A, b \\in B \\}$ denotes the sumset of $A$ and $B$. Clearly one has equality when $\\mu_G(A)+\\mu_G(B) \\geq 1$. Another way in which equality can be obtained is when $A = \\phi^{-1}(I), B = \\phi^{-1}(J)$ for some continuous surjective homomorphism $\\phi: G \\to {\\bf R}/{\\bf Z}$ and compact arcs $I,J \\subset {\\bf R}/{\\bf Z}$. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then $A,B$ are close to one of the above examples. We also give a more \"robust\" form of this theorem in which the sumset $A+B$ is replaced by the partial sumset $A +_\\varepsilon B :=\\{ 1_A * 1_B \\geq \\varepsilon \\}$ for some small $\\varepsilon >0$. In a subsequent paper with Joni Ter\\\"av\\\"ainen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-12T18:43:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "inverse theorem",
      "kemperman inequality",
      "compact connected abelian group",
      "probability haar measure",
      "compact subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}