{
  "id": "2104.03232",
  "version": "v1",
  "published": "2021-04-07T16:31:00.000Z",
  "updated": "2021-04-07T16:31:00.000Z",
  "title": "On small fractional parts of polynomial-like functions",
  "authors": [
    "Paolo Minelli"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for $f(x)=x^k+x^c$ where $k$ is a positive integer and $c>1$ is a non-integer, and any fixed $\\xi\\in [0,1]$ they obtained \\[\\min_{2\\leq p\\leq X} \\Vert \\xi \\lfloor f(p)\\rfloor \\Vert\\ll_{k,c,\\epsilon} X^{-\\rho_1(c,k)+\\epsilon}\\] for $\\rho_1(c,k)>0$ explicitly given. In the present note, we improve upon their results in the case $c>k$ and $c>4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-07T16:31:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J54",
      "11L07",
      "11L20"
    ],
    "keywords": [
      "small fractional parts",
      "polynomial-like functions",
      "tichy established diophantine inequalities",
      "positive integer",
      "non-integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}