{
  "id": "1603.06827",
  "version": "v1",
  "published": "2016-03-22T15:22:01.000Z",
  "updated": "2016-03-22T15:22:01.000Z",
  "title": "A new expander and improved bounds for $A(A+A)$",
  "authors": [
    "Oliver Roche-Newton"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\\{(a_1+a_2+a_3+a_4)^2+\\log a_5 :a_1,a_2,a_3,a_4,a_5 \\in A \\}| \\gg \\frac{|A|^2}{\\log |A|}.$$ This bound is optimal, up to logarithmic factors. The paper also gives new lower bounds for $|A(A-A)|$ and $|A(A+A)|$, improving on results from arXiv:1312.6438. The new bounds are $$|A(A-A)| \\gtrapprox |A|^{3/2+\\frac{1}{34}}$$ and $$|A(A+A)| \\gtrapprox |A|^{3/2+\\frac{5}{242}}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-22T15:22:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "main result",
      "paper concerns",
      "logarithmic factors",
      "lower bounds",
      "real numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160306827R"
    }
  }
}