{
  "id": "2501.13869",
  "version": "v1",
  "published": "2025-01-23T17:41:21.000Z",
  "updated": "2025-01-23T17:41:21.000Z",
  "title": "Unique continuation for locally uniformly distributed measures",
  "authors": [
    "Max Engelstein",
    "Ignasi Guillén-Mola"
  ],
  "categories": [
    "math.CA",
    "math.DG"
  ],
  "abstract": "In this note we show that the support of a locally $k$-uniform measure in $\\mathbb R^{n+1}$ satisfies a kind of unique continuation property. As a consequence, we show that locally uniformly distributed measures satisfy a weaker unique continuation property. This continues work of Kirchheim and Preiss (Math. Scand. 2002) and David, Kenig and Toro (Comm. Pure Appl. Math. 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J. Reine Angew. Math. 1987) that the support of a locally $n$-uniform measure in $\\mathbb R^{n+1}$ is contained in the zero set of a quadratic polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-23T17:41:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28C15",
      "58C35",
      "28A78",
      "49Q15"
    ],
    "keywords": [
      "locally uniformly distributed measures",
      "uniformly distributed measures satisfy",
      "uniform measure",
      "weaker unique continuation property",
      "lends additional evidence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}