{
  "id": "1202.2398",
  "version": "v3",
  "published": "2012-02-11T01:55:20.000Z",
  "updated": "2013-05-19T14:40:46.000Z",
  "title": "The Pompeiu Problem and Discrete Groups",
  "authors": [
    "Michael J. Puls"
  ],
  "comment": "Version two fixes some typos. Also a new section was added concerning the harmonicity of a function with regards to the mean-value property on two spheres. Corrected some more typos",
  "categories": [
    "math.FA",
    "math.GR"
  ],
  "abstract": "We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-19T14:40:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A15"
    ],
    "keywords": [
      "discrete group",
      "pompeiu problem",
      "nonabelian free group",
      "sufficient condition",
      "torsion free rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2398P"
    }
  }
}