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On points without universal expansions

Volume 184 / 2018

Karma Dajani, Kan Jiang Acta Arithmetica 184 (2018), 231-245 MSC: 11K55, 28D05, 28A80, 11A63. DOI: 10.4064/aa170323-16-2 Published online: 20 July 2018

Abstract

Let $1 \lt \beta \lt 2$. Given any $x\in[0, (\beta-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $\beta$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_n\beta^{-n}.$ For any $k\geq 1$ and any $(b_1\cdots b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1\cdots b_k$, then we call $(a_n)$ a universal $\beta$-expansion of $x$. Sidorov (2003) and Dajani and de Vries (2007) proved that for any $1 \lt \beta \lt 2$, Lebesgue almost every point has uncountably many universal $\beta$-expansions. In this paper we consider the set $V_{\beta}$ of points without universal $\beta$-expansions. For any $n\geq 2$, let $\beta_n$ be the $n$-bonacci number, i.e., $\beta^n=\beta^{n-1}+\beta^{n-2}+\cdots +\beta+1.$ Then $\dim_{H}(V_{\beta_n})=1$, where $\dim_{H}$ denotes the Hausdorff dimension. Similar results are available for some other algebraic numbers. As a corollary, we give some results on the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper [Nonlinearity 30 (2017)].

Authors

  • Karma DajaniDepartment of Mathematics
    Utrecht University
    Budapestlaan 6
    P.O. Box 80.000
    3508 TA Utrecht, The Netherlands
    e-mail
  • Kan JiangDepartment of Mathematics
    Ningbo University
    Ningbo, Zhejiang, People’s Republic of China
    e-mail

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