{
  "id": "2101.05797",
  "version": "v1",
  "published": "2021-01-14T18:58:48.000Z",
  "updated": "2021-01-14T18:58:48.000Z",
  "title": "Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals",
  "authors": [
    "Osama Khalil",
    "Manuel Luethi"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle thirds set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\\mathbb{R}^d$ (for any $d\\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the \"middle ninths Cantor set\"; i.e. the set of numbers whose base $9$ expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space $\\mathcal{L}_{d+1}$ of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of $S$-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on $\\mathcal{L}_{d+1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-14T18:58:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walks",
      "khintchines theorem",
      "spectral gap",
      "fractal measures",
      "mahler regarding cantors middle thirds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}