{
  "id": "2402.18297",
  "version": "v2",
  "published": "2024-02-28T12:42:33.000Z",
  "updated": "2024-09-07T09:08:40.000Z",
  "title": "Sums, Differences and Dilates",
  "authors": [
    "Jonathan Cutler",
    "Luke Pebody",
    "Amites Sarkar"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a set of integers $A$ and an integer $k$, write $A+k\\cdot A$ for the set $\\{a+kb:a\\in A,b\\in A\\}$. Hanson and Petridis showed that if $|A+A|\\le K|A|$ then $|A+2\\cdot A|\\le K^{2.95}|A|$. At a presentation of this result, Petridis stated that the highest known value for $\\frac{\\log(|A+2\\cdot A|/|A|)}{\\log(|A+A|/|A|)}$ (bounded above by 2.95) was $\\frac{\\log 4}{\\log 3}$. We show that, for all $\\epsilon>0$, there exist $A$ and $K$ with $|A+A|\\le K|A|$ but with $|A+2\\cdot A|\\ge K^{2-\\epsilon}|A|$. Further, we analyse a method of Ruzsa, and generalise it to give continuous analogues of the sizes of sumsets, differences and dilates. We apply this method to a construction of Hennecart, Robert and Yudin to prove that, for all $\\epsilon>0$, there exists a set $A$ with $|A-A|\\ge |A|^{2-\\epsilon}$ but with $|A+A|<|A|^{1.7354+\\epsilon}$. The second author would like to thank E. Papavassilopoulos for useful discussions about how to improve the efficiency of his computer searches.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-07T09:08:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differences",
      "second author",
      "computer searches",
      "generalise",
      "continuous analogues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}