{
  "id": "1403.7703",
  "version": "v2",
  "published": "2014-03-30T06:04:50.000Z",
  "updated": "2014-05-07T21:39:40.000Z",
  "title": "General systems of linear forms: equidistribution and true complexity",
  "authors": [
    "Hamed Hatami",
    "Pooya Hatami",
    "Shachar Lovett"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a near-equidistribution theorem that describes these distributions for the group $\\mathbb{F}_p^n$ when $p$ is a fixed prime. This fundamental fact is equivalent to a strong near-orthogonality statement regarding the higher-order characters, and was previously known only under various extra assumptions about the linear forms. As an application of our near-equidistribution theorem, we settle a conjecture of Gowers and Wolf on the true complexity of systems of linear forms for the group $\\mathbb{F}_p^n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-07T21:39:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear forms",
      "true complexity",
      "general systems",
      "near-equidistribution theorem",
      "higher-order fourier analysis examines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7703H"
    }
  }
}