{
  "id": "0711.3637",
  "version": "v1",
  "published": "2007-11-22T19:14:57.000Z",
  "updated": "2007-11-22T19:14:57.000Z",
  "title": "Uniformity seminorms on $\\ell^\\infty$ and applications",
  "authors": [
    "Bryna Kra",
    "Bernard Host"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\\Z/N\\Z$ introduced by Gowers in his proof of Szemer\\'edi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemer\\'edi's Theorem) defined by the authors. For each integer $k\\geq 1$, we define seminorms on $\\ell^\\infty(\\Z)$ analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-22T19:14:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A45"
    ],
    "keywords": [
      "uniformity seminorms",
      "applications",
      "weighted multiple ergodic convergence theorem",
      "nilsequence",
      "szemeredis theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.3637K"
    }
  }
}