{
  "id": "1701.01635",
  "version": "v1",
  "published": "2017-01-06T13:54:47.000Z",
  "updated": "2017-01-06T13:54:47.000Z",
  "title": "A Second Wave of Expanders over Finite Fields",
  "authors": [
    "Brendan Murphy",
    "Giorgis Petridis"
  ],
  "comment": "CANT (Combinatorial and Additive Number Theory) 2016",
  "categories": [
    "math.CO"
  ],
  "abstract": "This is an expository survey on recent sum-product results in finite fields. We present a number of sum-product or \"expander\" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly as large as possible, and if $|A|<p^{2/3}$ then the set in question is significantly larger that $A$. These results are based on a point-plane incidence bound of Rudnev, and are quantitatively stronger than a wave of earlier results following Bourgain, Katz, and Tao's breakthrough sum-product result. In addition, we present two geometric results: an incidence bound due to Stevens and de Zeeuw, and bound on collinear triples, and an example of an expander that breaks the threshold of $p^{2/3}$ required by the other results. We have simplified proofs wherever possible, and hope that this survey may serve as a compact guide to recent advances in arithmetic combinatorics over finite fields. We do not claim originality for any of the results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-06T13:54:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D99",
      "05B10",
      "11B30"
    ],
    "keywords": [
      "finite fields",
      "second wave",
      "taos breakthrough sum-product result",
      "point-plane incidence bound",
      "earlier results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}