{
  "id": "1803.02117",
  "version": "v1",
  "published": "2018-03-06T11:21:26.000Z",
  "updated": "2018-03-06T11:21:26.000Z",
  "title": "Equality case in van der Corput's inequality and collisions in multiple lattice tilings",
  "authors": [
    "Gennadiy Averkov"
  ],
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Van der Corput's provides the sharp bound vol(C) \\le m 2^d on the volume of a d-dimensional origin-symmetric convex body C that has 2m-1 points of the integer lattice in its interior. For m=1, a characterization of the equality case vol(C)= m 2^d is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for m \\ge 2, no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all m \\ge 2. Our result reveals that, the equality case for m \\ge 2 is more restrictive than for $m=1$. We also present consequences of our characterization in the context of multiple lattice tilings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-06T11:21:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C22",
      "52C07"
    ],
    "keywords": [
      "equality case",
      "van der corputs inequality",
      "multiple lattice tilings",
      "characterization",
      "d-dimensional origin-symmetric convex body"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}