{
  "id": "1306.3315",
  "version": "v2",
  "published": "2013-06-14T07:31:54.000Z",
  "updated": "2013-07-25T01:27:25.000Z",
  "title": "Metric number theory, lacunary series and systems of dilated functions",
  "authors": [
    "Christoph Aistleitner"
  ],
  "comment": "Survey paper for the RICAM workshop on \"Uniform Distribution and Quasi-Monte Carlo Methods\", held from October 14-18, 2013, in Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the \"Radon Series on Computational and Applied Mathematics\" by DeGruyter",
  "categories": [
    "math.NT",
    "math.CA",
    "math.HO"
  ],
  "abstract": "By a classical result of Weyl, for any increasing sequence $(n_k)_{k \\geq 1}$ of integers the sequence of fractional parts $(\\{n_k x\\})_{k \\geq 1}$ is uniformly distributed modulo 1 for almost all $x \\in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \\geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\\{n_k x\\})_{k \\geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \\geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \\geq q > 1, k \\geq 1$. In this case of quickly increasing $(n_k)_{k \\geq 1}$ the system $(\\{n_k x\\})_{k \\geq 1}$ (or, more general, $(f(n_k x))_{k \\geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \\emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena. Without any growth conditions on $(n_k)_{k \\geq 1}$ the situation becomes much more complicated, and the system $(f(n_k x))_{k \\geq 1}$ will typically fail to satisfy probabilistic limit theorems. An important problem which remains is to study the almost everywhere convergence of series $\\sum_{k=1}^\\infty c_k f(k x)$, which is closely related to finding upper bounds for maximal $L^2$-norms of the form $$ \\int_0^1 (\\max_{1 \\leq M \\leq N}| \\sum_{k=1}^M c_k f(kx)|^2 dx. $$ The most striking example of this connection is the equivalence of the Carleson convergence theorem and the Carleson--Hunt inequality for maximal partial sums of Fourier series. For general functions $f$ this is a very difficult problem, which is related to finding upper bounds for certain sums involving greatest common divisors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-25T01:27:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "11K38",
      "42A55",
      "60F15",
      "11A05",
      "42A20"
    ],
    "keywords": [
      "metric number theory",
      "lacunary series",
      "dilated functions",
      "finding upper bounds",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.3315A"
    }
  }
}