{
  "id": "1504.07837",
  "version": "v1",
  "published": "2015-04-29T12:48:14.000Z",
  "updated": "2015-04-29T12:48:14.000Z",
  "title": "Equidistribution of values of linear forms on a cubic hypersurface",
  "authors": [
    "Sam Chow"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $C$ be a cubic form with rational coefficients in $n$ variables, and let $h$ be the $h$-invariant of $C$. Let $L_1, \\ldots, L_r$ be linear forms with real coefficients such that if $\\boldsymbol{\\alpha} \\in \\mathbb{R}^r \\setminus \\{ \\boldsymbol{0} \\}$ then $\\boldsymbol{\\alpha} \\cdot \\mathbf{L}$ is not a rational form. Assume that $h > 16 + 8 r$. Let $\\boldsymbol{\\tau} \\in \\mathbb{R}^r$, and let $\\eta$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $\\mathbf{x} \\in [-P,P]^n$ to the system $C(\\mathbf{x}) = 0, \\: |\\mathbf{L}(\\mathbf{x}) - \\boldsymbol{\\tau}| < \\eta$. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the $h$-invariant condition with the hypothesis $n > 16 + 9 r$, and show that the system has an integer solution. Finally, we show that the values of $\\mathbf{L}$ at integer zeros of $C$ are equidistributed modulo one in $\\mathbb{R}^r$, requiring only that $h > 16$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-29T12:48:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D25",
      "11D75",
      "11J13",
      "11J71",
      "11P55"
    ],
    "keywords": [
      "linear forms",
      "cubic hypersurface",
      "equidistribution",
      "integer solution",
      "integer zeros"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150407837C"
    }
  }
}