{
  "id": "1211.4640",
  "version": "v1",
  "published": "2012-11-20T00:59:15.000Z",
  "updated": "2012-11-20T00:59:15.000Z",
  "title": "On a problem of Bourgain concerning the $L^1$-norm of exponential sums",
  "authors": [
    "Christoph Aistleitner"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Bourgain posed the problem of calculating $$ \\Sigma = \\sup_{n \\geq 1} ~\\sup_{k_1 <... < k_n} \\frac{1}{\\sqrt{n}}\\| \\sum_{j=1}^n e^{2 \\pi i k_j \\theta}\\|_{L^1([0,1])}. $$ It is clear that $\\Sigma \\leq 1$; beyond that, determining whether $\\Sigma < 1$ or $\\Sigma=1$ would have some interesting implications, for example concerning the problem whether all rank one transformations have singular maximal spectral type. In the present paper we prove $\\Sigma \\geq \\sqrt{\\pi}/2 \\approx 0.886$, by this means improving a result of Karatsuba. For the proof we use a quantitative two-dimensional version of the central limit theorem for lacunary trigonometric series, which in its original form is due to Salem and Zygmund.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-20T00:59:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A05",
      "33B10"
    ],
    "keywords": [
      "exponential sums",
      "bourgain concerning",
      "singular maximal spectral type",
      "central limit theorem",
      "lacunary trigonometric series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4640A"
    }
  }
}