{
  "id": "1502.03704",
  "version": "v1",
  "published": "2015-02-12T15:39:42.000Z",
  "updated": "2015-02-12T15:39:42.000Z",
  "title": "Improved bounds for arithmetic progressions in product sets",
  "authors": [
    "Dmitry Zhelezov"
  ],
  "comment": "To appear in Int. J. Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \\{bb'| \\, b, b' \\in B\\}$ cannot be greater than $O(n \\log n)$ which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers we improve the bound to $O_\\epsilon(n^{1 + \\epsilon})$ for arbitrary $\\epsilon > 0$ assuming the GRH.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-12T15:39:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25"
    ],
    "keywords": [
      "product set",
      "lower bound",
      "longest arithmetic progression",
      "natural numbers",
      "earlier paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}