{
  "id": "1805.00865",
  "version": "v1",
  "published": "2018-05-02T15:23:06.000Z",
  "updated": "2018-05-02T15:23:06.000Z",
  "title": "Sums of reciprocals of fractional parts",
  "authors": [
    "Reynold Fregoli"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\boldsymbol{\\alpha}\\in \\mathbb{R}^N$ and $Q\\geq 1$. We consider the sum $\\sum_{\\boldsymbol{q}\\in [-Q,Q]^N\\cap\\mathbb{Z}^N\\backslash\\{\\boldsymbol{0}\\}}\\|\\boldsymbol{\\alpha}\\cdot\\boldsymbol{q}\\|^{-1}$. Sharp upper bounds are known when $N=1$, using continued fractions or the three distance theorem. However, these techniques do not seem to apply in higher dimension. We introduce a different approach, based on a general counting result of Widmer for weakly admissible lattices, to establish sharp upper bounds for arbitrary $N$. Our result also sheds light on a question raised by L\\^{e} and Vaaler in 2013 on the sharpness of their lower bound $\\gg Q^N\\log Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-02T15:23:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional parts",
      "reciprocals",
      "establish sharp upper bounds",
      "distance theorem",
      "higher dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}