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Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

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Abstract

We consider the distribution of the orbits of the number 1 under the \(\beta \)-transformations \(T_\beta \) as \(\beta \) varies. Mainly, the size of the set of \(\beta >1\) for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set \(E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\Big \{\,\beta >1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \hbox { for infinitely many}, \, n\in \mathbb{N }\,\Big \}\) is determined, where \(x_0\) is a given point in \([0,1]\) and \(\{\ell _n\}_{n\ge 1}\) is a sequence of integers tending to infinity as \(n\rightarrow \infty \). For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of \(\beta \) with a common prefix in the expansion of 1) in the parameter space \(\{\,\beta \in \mathbb{R }: \beta >1\,\}\).

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Acknowledgments

This work was in part finished when some of the authors visited the Morning Center of Mathematics (Beijing). The authors are grateful for the host’s warm hospitality. This work was partially supported by NSFC Nos. 11126071, 10901066, 11225101 and 11171123.

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Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China

    Bing Li

  2. Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 , Oulu, Finland

    Bing Li

  3. Centre for Mathematical Sciences, Lund University, Box 118, 22100 , Lund, Sweden

    Tomas Persson

  4. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Baowei Wang & Jun Wu

Authors
  1. Bing Li
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  2. Tomas Persson
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  3. Baowei Wang
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  4. Jun Wu
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Correspondence to Baowei Wang.

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Li, B., Persson, T., Wang, B. et al. Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions. Math. Z. 276, 799–827 (2014). https://doi.org/10.1007/s00209-013-1223-0

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  • DOI: https://doi.org/10.1007/s00209-013-1223-0

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