{
  "id": "math/0607637",
  "version": "v1",
  "published": "2006-07-25T18:12:52.000Z",
  "updated": "2006-07-25T18:12:52.000Z",
  "title": "Multiple recurence and convergence for sequences related to the prime numbers",
  "authors": [
    "Nikos Frantzikinakis",
    "Bernard Host",
    "Bryna Kra"
  ],
  "comment": "14 pages. To appear in Crelle's Journal",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "For any measure preserving system $(X,\\mathcal{X},\\mu,T)$ and $A\\in\\mathcal{X}$ with $\\mu(A)>0$, we show that there exist infinitely many primes $p$ such that $\\mu\\bigl(A\\cap T^{-(p-1)}A\\cap T^{-2(p-1)}A\\bigr) > 0$ (the same holds with $p-1$ replaced by $p+1$). Furthermore, we show the existence of the limit in $L^2(\\mu)$ of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form $p-1$ (or $p+1$) for some prime $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-07-25T18:12:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "28D05"
    ],
    "keywords": [
      "prime numbers",
      "multiple recurence",
      "convergence",
      "positive upper density contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7637F"
    }
  }
}