{
  "id": "1703.09549",
  "version": "v1",
  "published": "2017-03-28T12:42:53.000Z",
  "updated": "2017-03-28T12:42:53.000Z",
  "title": "Variations on the sum-product problem II",
  "authors": [
    "Brendan Murphy",
    "Oliver Roche-Newton",
    "Ilya Shkredov"
  ],
  "comment": "This paper supercedes arXiv:1603.06827",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, as well as generalising a method therein to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \\subset \\mathbb R$: \\begin{align*} \\exists a \\in A \\text{ such that }|A(A+a)| &\\gtrsim |A|^{\\frac{3}{2}+\\frac{1}{186}}, |A(A-A)| &\\gtrsim |A|^{\\frac{3}{2}+\\frac{1}{34}}, |A(A+A)| &\\gtrsim |A|^{\\frac{3}{2}+\\frac{5}{242}}, |\\{(a_1+a_2+a_3+a_4)^2+\\log a_5 : a_i \\in A \\}| &\\gg \\frac{|A|^2}{\\log |A|}. \\end{align*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-28T12:42:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sum-product problem",
      "variations",
      "finite set",
      "main results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}