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Approximation orders of real numbers by \(\beta \)-expansions

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Abstract

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their \(\beta \)-expansions with the exponential order \(\beta ^{-n}\). Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under \(\beta \)-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of \(\beta \)-expansions.

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Acknowledgements

Thank the referee for helpful suggestions. The work was supported by NSFC 11771153, 11671151, 11801591, Guangdong Natural Science Foundation 2018B0303110005 and Fundamental Research Funds for the Central Universities SYSU-18lgpy65. The authors also thank Professor Christoph Bandt for the comments on the results.

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Authors and Affiliations

  1. School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China

    Lulu Fang

  2. School of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China

    Min Wu & Bing Li

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  1. Lulu Fang
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  2. Min Wu
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  3. Bing Li
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Correspondence to Bing Li.

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Fang, L., Wu, M. & Li, B. Approximation orders of real numbers by \(\beta \)-expansions. Math. Z. 296, 13–40 (2020). https://doi.org/10.1007/s00209-019-02402-w

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