{
  "id": "2211.08994",
  "version": "v1",
  "published": "2022-10-28T12:13:49.000Z",
  "updated": "2022-10-28T12:13:49.000Z",
  "title": "Product of difference sets of set of primes",
  "authors": [
    "Sayan Goswami"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "A. Fish proved that for any two positive density sets $A$ and $B$ in two measure preserving systems $\\left(X,\\mu,T\\right)$ and $\\left(Y,\\nu,S\\right)$, there exists $k\\in\\mathbb{Z}$ such that $k\\cdot \\mathbb{Z}\\subseteq R\\left(A\\right)\\cdot R\\left(B\\right)$, where $R\\left(A\\right)$ and $R\\left(B\\right)$ are the return times sets. As a consequence it follows that for any two sets of positive density $E_{1}$ and $E_{2}$ in $\\mathbb{Z}$, there exists $k\\in\\mathbb{Z}$ such that $k\\cdot \\mathbb{Z}\\subset\\left(E_{1}-E_{1}\\right)\\cdot\\left(E_{2}-E_{2}\\right).$ In this article we will show that the result is still true under sufficiently weak assumption. As a consequence we show that there exists $k\\in\\mathbb{N}$ such that $k\\cdot \\mathbb{N}\\subseteq\\left(\\mathbb{P}-\\mathbb{P}\\right)\\cdot\\left(\\mathbb{P}-\\mathbb{P}\\right)$, where $\\mathbb{P}$ is the set of primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-28T12:13:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "difference sets",
      "return times sets",
      "positive density sets",
      "consequence",
      "sufficiently weak assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}