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

  • Lulu Fang
  • Min Wu
  • Bing LiEmail author


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.


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

Mathematics Subject Classification

Primary 11K55 28A80 Secondary 37B10 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-Sen UniversityGuangzhouPeople’s Republic of China
  2. 2.School of MathematicsSouth China University of TechnologyGuangzhouPeople’s Republic of China

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