{
  "id": "1504.04904",
  "version": "v1",
  "published": "2015-04-20T00:50:05.000Z",
  "updated": "2015-04-20T00:50:05.000Z",
  "title": "Difference Sets and Sums of Polynomials",
  "authors": [
    "Neil Lyall",
    "Alex Rice"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NT",
    "math.CA",
    "math.CO"
  ],
  "abstract": "We provide upper bounds on the largest subsets of $\\{1,2,\\dots,N\\}$ with no differences of the form $h_1(n_1)+\\cdots+h_{\\ell}(n_{\\ell})$ with $n_i\\in \\mathbb{N}$ or $h_1(p_1)+\\cdots+h_{\\ell}(p_{\\ell})$ with $p_i$ prime, where $h_i\\in \\mathbb{Z}[x]$ lie in the largest possible classes of polynomials. For example, we show that a subset of $\\{1,2,\\dots,N\\}$ free of nonzero differences of the form $n^j+m^k$ for fixed $j,k\\in \\mathbb{N}$ has density at most $e^{-c(\\log N)^{1/4}}$ for some $c=c(j,k)>0$. Our results, obtained by adapting two Fourier analytic, circle method-driven strategies, either recover or improve upon all previous results for a single polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-20T00:50:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "difference sets",
      "circle method-driven strategies",
      "single polynomial",
      "largest subsets",
      "nonzero differences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150404904L"
    }
  }
}