{
  "id": "1303.5001",
  "version": "v3",
  "published": "2013-03-19T19:54:50.000Z",
  "updated": "2014-10-22T09:18:00.000Z",
  "title": "Measurable events indexed by words",
  "authors": [
    "Pandelis Dodos",
    "Vassilis Kanellopoulos",
    "Konstantinos Tyros"
  ],
  "comment": "49 pages, no figures. This article is a sequel to arXiv:1209.4985",
  "journal": "Journal of Combinatorial Theory, Series A 127 (2014), 176-223",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "For every integer $k\\geq 2$ let $[k]^{<\\mathbb{N}}$ be the set of all words over $k$, that is, all finite sequences having values in $[k]:=\\{1,...,k\\}$. A Carlson-Simpson tree of $[k]^{<\\mathbb{N}}$ of dimension $m\\geq 1$ is a subset of $[k]^{<\\mathbb{N}}$ of the form \\[ \\{w\\}\\cup \\big\\{w^{\\smallfrown}w_0(a_0)^{\\smallfrown}...^{\\smallfrown}w_{n}(a_n): n\\in \\{0,...,m-1\\} \\text{ and } a_0,...,a_n\\in [k]\\big\\} \\] where $w$ is a word over $k$ and $(w_n)_{n=0}^{m-1}$ is a finite sequence of left variable words over $k$. We study the behavior of a family of measurable events in a probability space indexed by the elements of a Carlson-Simpson tree of sufficiently large dimension. Specifically we show the following. For every integer $k\\geq 2$, every $0<\\varepsilon\\leq 1$ and every integer $n\\geq 1$ there exists a strictly positive constant $\\theta(k,\\varepsilon,n)$ with the following property. If $m$ is a given positive integer, then there exists an integer $\\mathrm{Cor}(k,\\varepsilon,m)$ such that for every Carlson--Simpson tree $T$ of $[k]^{<\\mathbb{N}}$ of dimension at least $\\mathrm{Cor}(k,\\varepsilon,m)$ and every family $\\{A_t:t\\in T\\}$ of measurable events in a probability space $(\\Omega,\\Sigma,\\mu)$ satisfying $\\mu(A_t)\\geq \\varepsilon$ for every $t\\in T$, there exists a Carlson--Simpson tree $S$ of dimension $m$ with $S\\subseteq T$ and such that for every nonempty $F\\subseteq S$ we have \\[\\mu\\Big(\\bigcap_{t\\in F} A_t\\Big) \\geq \\theta(k,\\varepsilon,|F|). \\] The proof is based, among others, on the density version of the Carlson--Simpson Theorem established recently by the authors, as well as, on a partition result -- of independent interest -- closely related to the work of T. J. Carlson, and H. Furstenberg and Y. Katznelson. The argument is effective and yields explicit lower bounds for the constants $\\theta(k,\\varepsilon,n)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-13T19:39:11.000Z",
      "abstract": "For every integer $k\\meg 2$ let $[k]^{<\\nn}$ be the set of all \\textit{words} over $k$, that is, all finite sequences having values in [k]:=\\{1,...,k\\}$. A \\textit{Carlson-Simpson tree} of $[k]^{<\\nn}$ of dimension $m\\meg 1$ is a subset of $[k]^{<\\nn}$ of the form \\[ \\{w\\}\\cup \\big\\{w^{\\con}w_0(a_0)^{\\con}...^{\\con}w_{n}(a_n): n\\in \\{0,...,m-1\\} \\text{and} a_0,...,a_n\\in [k]\\big\\} \\] where $w$ is a word over $k$ and $(w_n)_{n=0}^{m-1}$ is a finite sequence of left variable words over $k$. We study the behavior of a family of measurable events in a probability space indexed by the elements of a Carlson-Simpson tree of sufficiently large dimension. Specifically we show the following. \\medskip \\noindent \\textit{For every integer $k\\meg 2$, every $0<\\ee\\mik 1$ and every integer $n\\meg 1$ there exists a strictly positive constant $\\theta(k,\\ee,n)$ with the following property. If $m$ is a given positive integer, then there exists an integer $\\mathrm{Cor}(k,\\ee,m)$ such that for every Carlson--Simpson tree $T$ of $[k]^{<\\nn}$ of dimension at least $\\mathrm{Cor}(k,\\ee,m)$ and every family $\\{A_t:t\\in T\\}$ of measurable events in a probability space $(\\Omega,\\Sigma,\\mu)$ satisfying $\\mu(A_t)\\meg \\ee$ for every $t\\in T$, there exists a Carlson--Simpson tree $S$ of dimension $m$ with $S\\subseteq T$ and such that for every nonempty $F\\subseteq S$ we have} \\[\\mu\\Big(\\bigcap_{t\\in F} A_t\\Big) \\meg \\theta(k,\\ee,|F|). \\] The proof is based, among others, on the density version of the Carlson--Simpson Theorem established recently by the authors, as well as, on a refinement -- of independent interest -- of a partition result due to H. Furstenberg and Y. Katznelson. The argument is effective and yields explicit lower bounds for the constants $\\theta(k,\\ee,n)$.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-22T09:18:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measurable events",
      "finite sequences"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.5001D"
    }
  }
}