{
  "id": "1812.01981",
  "version": "v1",
  "published": "2018-12-05T13:18:46.000Z",
  "updated": "2018-12-05T13:18:46.000Z",
  "title": "On Products of Shifts in Arbitrary Fields",
  "authors": [
    "Audie Warren"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field $\\mathbb{F}$, for all $A \\subset \\mathbb{F}$ finite with $|A| < p^{1/4},$ we have $$|A(A+1)| \\gtrsim |A|^{11/9}, \\qquad |AA| + |(A+1)(A+1)| \\gtrsim |A|^{11/9}.$$ This improves upon the exponent of $6/5$ given by an incidence theorem of Stevens and de Zeeuw.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-05T13:18:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary field",
      "incidence theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}