{
  "id": "2012.11686",
  "version": "v1",
  "published": "2020-12-21T21:20:34.000Z",
  "updated": "2020-12-21T21:20:34.000Z",
  "title": "A Polynomial Roth Theorem for Corners in Finite Fields",
  "authors": [
    "Rui Han",
    "Michael T Lacey",
    "Fan Yang"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CA",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We prove a Roth type theorem for polynomial corners in the finite field setting. Let $\\phi_1$ and $\\phi_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \\subset \\mathbb F_p \\times \\mathbb F_p$ with $ \\lvert A\\rvert > p ^{2 - \\frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + \\phi_1 (y), x_2), (x_1, x_2 + \\phi_2 (y))$. The study of these questions on $ \\mathbb F_p$ was started by Bourgain and Chang. Our Theorem adapts the argument of Dong, Li and Sawin, in particular relying upon deep Weil type inequalities established by N. Katz.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-21T21:20:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial roth theorem",
      "finite field",
      "deep weil type inequalities",
      "roth type theorem",
      "theorem adapts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}