{
  "id": "1403.1629",
  "version": "v1",
  "published": "2014-03-07T01:19:47.000Z",
  "updated": "2014-03-07T01:19:47.000Z",
  "title": "On the law of the iterated logarithm for trigonometric series with bounded gaps II",
  "authors": [
    "Christoph Aistleitner",
    "Katusi Fukuyama"
  ],
  "comment": "This manuscript is a complement to the paper \"On the law of the iterated logarithm for trigonometric series with bounded gaps\", Probab. Th. Rel. Fields, 154 (2012), no. 3-4, 607--620, by the same authors",
  "categories": [
    "math.NT",
    "math.CA",
    "math.PR"
  ],
  "abstract": "It is well-known that for a quickly increasing sequence $(n_k)_{k \\geq 1}$ the functions $(\\cos 2 \\pi n_k x)_{k \\geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k \\geq 1}$ is relaxed then this almost-independent behavior generally fails. Still, probabilistic constructions show that for \\emph{some} very slowly increasing sequences $(n_k)_{k \\geq 1}$ this almost-independence property is preserved. For example, there exists $(n_k)_{k \\geq 1}$ having bounded gaps such that the normalized sums $\\sum \\cos 2 \\pi n_k x$ satisfy the central limit theorem (CLT). However, due to a ``loss of mass'' phenomenon the variance in the CLT for a sequence with bounded gaps is always smaller than $1/2$. In the case of the law of the iterated logarithm (LIL) the situation is different; as we proved in an earlier paper, there exists $(n_k)_{k \\geq 1}$ with bounded gaps such that $$ \\limsup_{N \\to \\infty} \\frac{\\left| \\sum_{k=1}^N \\cos 2 \\pi n_k x \\right|}{\\sqrt{N \\log \\log N}} = \\infty \\qquad \\textrm{for almost all $x$.} $$ In the present paper we prove a complementary results showing that any prescribed limsup-behavior in the LIL is possible for sequences with bounded gaps. More precisely, we show that for any real number $\\Lambda \\geq 0$ there exists a sequence of integers $(n_k)_{k \\geq 1}$ satisfying $n_{k+1} - n_{k} \\in \\{1,2\\}$ such that the limsup in the LIL equals $\\Lambda$ for almost all $x$. Similar results are proved for sums $\\sum f(n_k x)$ and for the discrepancy of $(\\langle n_k x \\rangle)_{k \\geq 1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-07T01:19:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "11K38",
      "42A32"
    ],
    "keywords": [
      "bounded gaps",
      "iterated logarithm",
      "trigonometric series",
      "increasing sequence",
      "independent random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.1629A"
    }
  }
}