{
  "id": "2009.04229",
  "version": "v1",
  "published": "2020-09-09T11:34:18.000Z",
  "updated": "2020-09-09T11:34:18.000Z",
  "title": "Densities on Dedekind domains, completions and Haar measure",
  "authors": [
    "Luca Demangos",
    "Ignazio Longhi"
  ],
  "comment": "34 pages, no figures. This is a preliminary version",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $D$ be the ring of $S$-integers in a global field and $\\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is equal to the Haar measure of its closure in $\\hat{D}$. In order to have a precise statement, we give a general definition of density which encompasses the most commonly used ones. Using it we provide a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. In another direction, we extend the Davenport-Erd\\H{o}s theorem to every $D$ as above and offer a new interpretation of it as a \"density=measure\" result. Our point of view also provides a simple proof that in any $D$ the set of elements divisible by at most $k$ distinct primes has density 0 for any natural number $k$. Finally, we show that the group of units of $\\hat{D}$ is contained in the closure of the set of irreducible elements of $D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-09T11:34:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "haar measure",
      "dedekind domains",
      "global field",
      "profinite completion",
      "sufficient condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}