{
  "id": "1810.03680",
  "version": "v1",
  "published": "2018-10-08T20:00:04.000Z",
  "updated": "2018-10-08T20:00:04.000Z",
  "title": "Binary Quadratic Forms in Difference Sets",
  "authors": [
    "Alex Rice"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT",
    "math.CA",
    "math.CO"
  ],
  "abstract": "We show that if $h(x,y)=ax^2+bxy+cy^2\\in \\mathbb{Z}[x,y]$ satisfies $b^2\\neq 4ac$, then any subset of $\\{1,2,\\dots,N\\}$ with no nonzero differences in the image of $h$ has size at most a constant depending on $h$ times $N\\exp(-c\\sqrt{\\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an $L^2$ density increment strategy previously used to establish analogous results for sums of one or more single-variable polynomials. Our exposition is thorough and self-contained, in order to serve as an accessible gateway for readers who are unfamiliar with previous implementations of these techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-08T20:00:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binary quadratic forms",
      "difference sets",
      "density increment strategy",
      "nonzero differences",
      "establish analogous results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}