{
  "id": "1907.13031",
  "version": "v1",
  "published": "2019-07-30T15:44:49.000Z",
  "updated": "2019-07-30T15:44:49.000Z",
  "title": "Dimension theory of Diophantine approximation related to $β$-transformations",
  "authors": [
    "Wanlou WU",
    "Lixuan Zheng"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $T_\\beta$ be the $\\beta$-transformation on $[0,1)$ defined by $$T_\\beta(x)=\\beta x\\text{ mod }1.$$ We study the Diophantine approximation of the orbit of a point $x$ under $T_\\beta$. Precisely, for given two positive functions $\\psi_1,~\\psi_2:\\mathbb{N}\\rightarrow\\mathbb{R}^+$, define $$\\mathcal{L}(\\psi_1):=\\left\\{x\\in[0,1]:T_\\beta^n x<\\psi_1(n),\\text{ for infinitely many $n\\in\\mathbb{N}$}\\right\\},$$ $$\\mathcal{U}(\\psi_2):=\\left\\{x\\in [0,1]:\\forall~N\\gg1,~\\exists~n\\in[0,N],~s.t.~T^n_\\beta x<\\psi_2(N)\\right\\},$$ where $\\gg$ means large enough. We compute the Hausdorff dimension of the set $\\mathcal{L}(\\psi_1)\\cap\\mathcal{U}(\\psi_2)$. As a corollary, we estimate the Hausdorff dimension of the set $\\mathcal{U}(\\psi_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T15:44:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "dimension theory",
      "transformation",
      "hausdorff dimension",
      "means large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}