{
  "id": "1609.03631",
  "version": "v1",
  "published": "2016-09-12T23:10:24.000Z",
  "updated": "2016-09-12T23:10:24.000Z",
  "title": "A spectral refinement of the Bergelson-Host-Kra decomposition and new multiple ergodic theorems",
  "authors": [
    "Joel Moreira",
    "Florian Karl Richter"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate how spectral properties of a measure preserving system $(X,\\mathcal{B},\\mu,T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\\mathbb{N}\\to\\mathbb{N}$ we provide natural conditions on the spectrum $\\sigma(T)$ such that for all $f_1,\\ldots,f_k\\in L^\\infty$, \\begin{equation*} \\lim_{N\\to\\infty} \\frac{1}{N} \\sum_{n=1}^N \\prod_{j=1}^k T^{ja(n)}f_j = \\lim_{N\\rightarrow\\infty} \\frac{1}{N} \\sum_{n=1}^N \\prod_{j=1}^k T^{jn}f_j \\end{equation*} in $L^2$-norm. In particular, our results apply to infinite arithmetic progressions $a(n)=qn+r$, Beatty sequences $a(n)=\\lfloor \\theta n+\\gamma\\rfloor$, the sequence of squarefree numbers $a(n)=q_n$, and the sequence of prime numbers $a(n)=p_n$. From our main results we derive that the set of prime numbers is a set of multiple recurrence for totally ergodic systems. We also obtain new refinements of Szemer{\\'e}di's theorem via Furstenberg's correspondence principle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-12T23:10:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37A30",
      "37A45",
      "05D10"
    ],
    "keywords": [
      "multiple ergodic theorems",
      "bergelson-host-kra decomposition",
      "spectral refinement",
      "prime numbers",
      "infinite arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}