{
  "id": "1011.4600",
  "version": "v3",
  "published": "2010-11-20T19:28:26.000Z",
  "updated": "2011-03-24T04:42:28.000Z",
  "title": "Higher-order Fourier analysis of $\\mathbb{F}_p^n$ and the complexity of systems of linear forms",
  "authors": [
    "Hamed Hatami",
    "Shachar Lovett"
  ],
  "comment": "final version, 25 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Consider a subset $A$ of $\\mathbb{F}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degree of uniformity $k$ such that assuming that $\\|f_2\\|_{U^k}$ is sufficiently small, it is possible to discard $f_2$ and replace $1_A$ with $f_1$ in the average over this family of linear forms, affecting it only negligibly. Previously, Gowers and Wolf solved this problem for the case where $f_1$ is a constant function. Furthermore, our main result solves Problem 7.6 in [W. T. Gowers and J. Wolf. Linear forms and higher-degree uniformity for functions on $\\mathbb{F}_p^n$. Geom. Funct. Anal., 21(1):36--69, 2011] regarding the analytic averages that involve more than one subset of $\\mathbb{F}_p^n$.] regarding the analytic averages that involve more than one subset of $\\mathbb{F}_p^n$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-03-24T04:42:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11T24"
    ],
    "keywords": [
      "linear forms",
      "higher-order fourier analysis",
      "analytic averages",
      "complexity",
      "higher-degree uniformity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.4600H"
    }
  }
}