{
  "id": "1802.02836",
  "version": "v1",
  "published": "2018-02-08T12:50:36.000Z",
  "updated": "2018-02-08T12:50:36.000Z",
  "title": "Convolutions of sets with bounded VC-dimension are uniformly continuous",
  "authors": [
    "Olof Sisask"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "abstract": "We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\\{ A \\cap(xA) : x \\in A\\cdot A^{-1} \\}$. We show that if a finite subset $A$ of an abelian group has bounded VC-dimension, then the convolution $1_A*1_{-A}$ is Bohr uniformly continuous, in a quantitatively strong sense. This generalises and strengthens a version of the stable arithmetic regularity lemma of Terry and Wolf in various ways. In particular, it directly implies that the Polynomial Bogolyubov--Ruzsa Conjecture --- a strong version of the Polynomial Freiman--Ruzsa Conjecture --- holds for sets with bounded VC-dimension. We also prove some results in the non-abelian setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-08T12:50:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded vc-dimension",
      "uniformly continuous",
      "convolution",
      "polynomial bogolyubov-ruzsa conjecture",
      "polynomial freiman-ruzsa conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}