{
  "id": "1810.11110",
  "version": "v1",
  "published": "2018-10-25T21:14:26.000Z",
  "updated": "2018-10-25T21:14:26.000Z",
  "title": "On the optimal rate of equidistribution in number fields",
  "authors": [
    "Mikolaj Fraczyk",
    "Anna Szumowicz"
  ],
  "comment": "41 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k$ be a number field. We study how well can finite sets of $\\mathcal O_k$ equidistribute modulo powers of prime ideals, for all prime ideals at the same time. Our main result states that the optimal rate of equidistribution in $\\mathcal O_k$ predicted by the local contstraints cannot be achieved unless $k=\\mathcal Q$. We deduce that $\\mathcal Q$ is the only number field where the ring of integers $\\mathcal O_k$ admits a simultaneous $\\frak p$-ordering, answering a question of Bhargava. Along the way we establish a non-trivial upper bound on the number of solutions $x\\in \\mathcal O_k$ of the inequality $|N_{k/\\mathcal Q}(x(a-x))|\\leq X^2$ where $X$ is a positive real parameter and $a\\in\\mathcal O_k$ is of norm at least $e^{-B}X$ for a fixed real number $B$. The latter can be translated as an upper bound on the average number of solutions of certain unit equations in $\\mathcal O_k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-25T21:14:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11K38",
      "13F20",
      "11D57"
    ],
    "keywords": [
      "number field",
      "optimal rate",
      "equidistribution",
      "prime ideals",
      "main result states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}