{
  "id": "0705.0715",
  "version": "v1",
  "published": "2007-05-04T22:49:01.000Z",
  "updated": "2007-05-04T22:49:01.000Z",
  "title": "Sum-product estimates via directed expanders",
  "authors": [
    "Van Vu"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\\F_q[x_1, x_2]$. For a set $A \\subset \\F_q$, define $P(A):=\\{P(x_1, x_2) | x_i \\in A \\}$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the following holds \\vskip2mm \\centerline{\\it If $|A+A|$ is small, then $|P(A)|$ is large.} \\vskip2mm The case $P=x_1x_2$ corresponds to the well-known sum-product problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-04T22:49:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sum-product estimates",
      "directed expanders",
      "well-known sum-product problem",
      "polynomial",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.0715V"
    }
  }
}