{
  "id": "1801.09591",
  "version": "v1",
  "published": "2018-01-29T15:56:40.000Z",
  "updated": "2018-01-29T15:56:40.000Z",
  "title": "A note on expansion in prime fields",
  "authors": [
    "Tuomas Orponen",
    "Laura Venieri"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $\\beta,\\epsilon \\in (0,1]$, and $k \\geq \\exp(122 \\max\\{1/\\beta,1/\\epsilon\\})$. We prove that if $A,B$ are subsets of a prime field $\\mathbb{Z}_{p}$, and $|B| \\geq p^{\\beta}$, then there exists a sum of the form $$S = a_{1}B \\pm \\ldots \\pm a_{k}B, \\qquad a_{1},\\ldots,a_{k} \\in A,$$ with $|S| \\geq 2^{-12}p^{-\\epsilon}\\min\\{|A||B|,p\\}$. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $\\alpha < 1$ and $\\beta,\\delta > 0$, there exists $\\epsilon > 0$ such that the following holds. If $A,B,E \\subset \\mathbb{Z}_{p}$ satisfy $|A| \\leq p^{\\alpha}$, $|B| \\geq p^{\\beta}$, and $|B||E| \\geq p^{\\delta}|A|$, then there exists $t \\in E$ such that $$|A + tB| \\geq c p^{\\epsilon}|A|,$$ for some absolute constant $c > 0$. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-29T15:56:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11B13"
    ],
    "keywords": [
      "prime field",
      "polynomial method",
      "sharper estimate",
      "absolute constant",
      "sum-product estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}