{
  "id": "1510.03645",
  "version": "v1",
  "published": "2015-10-13T12:19:11.000Z",
  "updated": "2015-10-13T12:19:11.000Z",
  "title": "Rotations by roots of unity and Diophantine approximation",
  "authors": [
    "Romanos-Diogenes Malikiosis"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a fixed integer $n$, we study the question whether at least one of the numbers $\\Re X\\omega^k$, $1\\leq k\\leq n$, is $\\varepsilon$-close to an integer, for any possible value of $X\\in\\mathbb{C}$, where $\\omega$ is a primitive $n$th root of unity. It turns out that there is always a $X$ for which the above numbers are concentrated around $1/2\\bmod1$. The distance from $1/2$ depends only on the local properties of $n$, rather than its magnitude. This is directly related the so-called \"pyjama\" problem which was solved recently.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-13T12:19:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J71"
    ],
    "keywords": [
      "diophantine approximation",
      "th root",
      "local properties",
      "fixed integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003645M"
    }
  }
}