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

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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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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Keywords

  • Approximation order
  • \(\beta \)-Expansions
  • Hausdorff dimension

Mathematics Subject Classification

  • Primary 11K55
  • 28A80
  • Secondary 37B10