{
  "id": "1409.4258",
  "version": "v1",
  "published": "2014-09-12T16:30:16.000Z",
  "updated": "2014-09-12T16:30:16.000Z",
  "title": "Sums of cubes 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 cubes $F(x_1,\\ldots,x_s) = (x_1 - \\mu_1)^3 + \\ldots + (x_s - \\mu_s)^3$. We show that if $\\eta$ is real, $\\tau >0$ is sufficiently large, and $s \\ge 9$, then there exist integers $x_1 > \\mu_1, \\ldots, x_s > \\mu_s$ such that $|F(\\mathbf{x})- \\tau| < \\eta$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s \\ge 5$. For $s \\ge 11$, we provide an asymptotic formula. If $s \\ge 6$ then $F(\\mathbf{Z}^s)$ is dense on the reals. Given nine variables, we can generalise this to sums of univariate cubic polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-12T16:30:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D75",
      "11E76",
      "11P05"
    ],
    "keywords": [
      "univariate cubic polynomials",
      "full density result",
      "asymptotic formula",
      "real numbers",
      "real analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4258C"
    }
  }
}