{
  "id": "1910.01437",
  "version": "v1",
  "published": "2019-10-03T13:04:29.000Z",
  "updated": "2019-10-03T13:04:29.000Z",
  "title": "On the pair correlations of powers of real numbers",
  "authors": [
    "Christoph Aistleitner",
    "Simon Baker"
  ],
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-03T13:04:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "11K60"
    ],
    "keywords": [
      "real numbers",
      "martingale approximation method",
      "extend koksmas theorem",
      "fractional parts",
      "koksma states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}