{
  "id": "1704.02909",
  "version": "v1",
  "published": "2017-04-10T15:35:10.000Z",
  "updated": "2017-04-10T15:35:10.000Z",
  "title": "Fourier dimension and spectral gaps for hyperbolic surfaces",
  "authors": [
    "Jean Bourgain",
    "Semyon Dyatlov"
  ],
  "comment": "27 pages, 3 figures",
  "categories": [
    "math.CA",
    "math.AP",
    "math.CO",
    "math.DS"
  ],
  "abstract": "We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=\\Gamma\\backslash\\mathbb H^2$ which depends only on the dimension $\\delta$ of the limit set. More precisely, we show that when $\\delta>0$ there exists $\\varepsilon_0=\\varepsilon_0(\\delta)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\\Re s>\\delta-\\varepsilon_0$. The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl. The key new component is a Fourier decay bound for the Patterson-Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group $\\Gamma$ are nonlinear, together with estimates on exponential sums due to Bourgain which follow from the discretized sum-product theorem in $\\mathbb R$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-10T15:35:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier dimension",
      "fractal uncertainty principle approach",
      "convex co-compact hyperbolic surface",
      "selberg zeta function",
      "essential spectral gap"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}