{
  "id": "1407.4736",
  "version": "v2",
  "published": "2014-07-17T16:57:16.000Z",
  "updated": "2014-12-01T18:17:53.000Z",
  "title": "(Uniform) Convergence of Twisted Ergodic Averages",
  "authors": [
    "Tanja Eisner",
    "Ben Krause"
  ],
  "comment": "31 pages, the referee's suggestions incorporated, references added, typos corrected. A uniform estimate of the ergodic averages with Hardy field weights by the corresponding Gowers-Host-Kra uniformity seminorms is added, see Theorem 2.11. To appear in Ergodic Theory Dynam. Systems",
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\\Sigma,\\mu)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field functions, $p$: \\[ \\{\\frac{1}{N} \\sum_{n\\leq N} e(p(n)) T^{n}f(x) \\} \\] and for \"twisted\" polynomial ergodic averages: \\[ \\{\\frac{1}{N} \\sum_{n\\leq N} e(n \\theta) T^{P(n)}f(x) \\} \\] for certain classes of badly approximable $\\theta \\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $\\mu$-a.e. for $f \\in L^p(X), \\ p >1,$ and arbitrary $\\theta \\in [0,1]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-17T16:57:16.000Z",
      "abstract": "Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\\Sigma,\\mu)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field functions, $p$: \\[ \\left\\{ \\frac{1}{N} \\sum_{n\\leq N} e( p(n) ) T^{n}f(x) \\right\\} \\] and for \"twisted\" polynomial ergodic averages: \\[ \\left\\{ \\frac{1}{N} \\sum_{n\\leq N} e(n \\theta) T^{P(n)}f(x) \\right\\} \\] for certain classes of badly approximable $\\theta \\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $\\mu$-a.e.\\ for $f \\in L^p(X), \\ p >1,$ and arbitrary $\\theta \\in [0,1]$.",
      "comment": "28 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-01T18:17:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "twisted ergodic averages",
      "convergence",
      "non-atomic probability space",
      "hardy field functions",
      "polynomial ergodic averages"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.4736E"
    }
  }
}