{
  "id": "1409.4259",
  "version": "v1",
  "published": "2014-09-12T16:43:35.000Z",
  "updated": "2014-09-12T16:43:35.000Z",
  "title": "Waring's problem with shifts",
  "authors": [
    "Sam Chow"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mu_1, \\ldots, \\mu_s$ be real numbers, with $\\mu_1$ irrational. We investigate sums of shifted $k$th powers $\\mathfrak{F}(x_1, \\ldots, x_s) = (x_1 - \\mu_1)^k + \\ldots + (x_s - \\mu_s)^k$. For $k \\ge 4$, we bound the number of variables needed to ensure that if $\\eta$ is real and $\\tau > 0$ is sufficiently large then there exist integers $x_1 > \\mu_1, \\ldots, x_s > \\mu_s$ such that $|\\mathfrak{F}(\\mathbf{x}) - \\tau| < \\eta$. This is a real analogue to Waring's problem. When $s \\ge 2k^2-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree $k$ polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-12T16:43:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D75",
      "11E76",
      "11P05"
    ],
    "keywords": [
      "warings problem",
      "general univariate degree",
      "th powers",
      "real numbers",
      "real analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4259C"
    }
  }
}