{
  "id": "1612.09040",
  "version": "v1",
  "published": "2016-12-29T05:32:40.000Z",
  "updated": "2016-12-29T05:32:40.000Z",
  "title": "Spectral gaps without the pressure condition",
  "authors": [
    "Jean Bourgain",
    "Semyon Dyatlov"
  ],
  "comment": "35 pages, 5 figures",
  "categories": [
    "math.CA",
    "math.AP",
    "math.DS",
    "math.SP",
    "nlin.CD"
  ],
  "abstract": "For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $\\delta$ of the limit set, in particular we do not require the pressure condition $\\delta\\leq {1\\over 2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed in Dyatlov-Zahl. The main new ingredient is the fractal uncertainty principle for $\\delta$-regular sets with $\\delta<1$, which may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-29T05:32:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pressure condition",
      "convex co-compact hyperbolic surfaces",
      "essential spectral gap",
      "selberg zeta function",
      "fractal uncertainty principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}