Advertisement

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

  • Lulu Fang
  • Min Wu
  • Bing LiEmail author
Article
  • 13 Downloads

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.

Keywords

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

Mathematics Subject Classification

Primary 11K55 28A80 Secondary 37B10 

Notes

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.

References

  1. 1.
    Aaronson, J., Nakada, H.: On the mixing coefficients of piecewise monotonic maps. Isr. J. Math. 148, 1–10 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adamczewski, B., Bugeaud, Y.: Dynamics for \(\beta \)-shifts and Diophantine approximation. Ergod. Theory Dyn. Syst. 27(6), 1695–1711 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ban, J.-C., Li, B.: The multifractal spectra for the recurrence rates of beta-transformations. J. Math. Anal. Appl. 420(2), 1662–1679 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barreira, L., Iommi, G.: Frequency of digits in the Lüroth expansion. J. Number Theory 129(6), 1479–1490 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blanchard, F.: \(\beta \)-expansions and symbolic dynamics. Theor. Comput. Sci. 65(2), 131–141 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bradley, R.: Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2, 107–144 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bugeaud, Y., Wang, B.-W.: Distribution of full cylinders and the Diophantine properties of the orbits in \(\beta \)-expansions. J. Fractal Geom. 1(2), 221–241 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Buzzi, J.: Specification on the interval. Trans. Am. Math. Soc. 349(7), 2737–2754 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dajani, K., Kraaikamp, C.: On approximation by Lüroth series. J. Théor. Nombres Bordeaux 8(2), 331–346 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)zbMATHGoogle Scholar
  11. 11.
    Fan, A.-H., Wu, J.: Approximation orders of formal Laurent series by Oppenheim rational functions. J. Approx. Theory 121(2), 269–286 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fan, A.-H., Wu, J.: Metric properties and exceptional sets of the Oppenheim expansions over the field of Laurent series. Constr. Approx. 20(4), 465–495 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fan, A., Liao, L., Wang, B., Wu, J.: On Khintchine exponents and Lyapunov exponents of continued fractions. Ergod. Theory Dyn. Syst. 29(1), 73–109 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fan, A.-H., Wang, B.-W.: On the lengths of basic intervals in beta expansions. Nonlinearity 25(5), 1329–1343 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fang, L., Wu, M., Li, B.: Limit theorems related to beta-expansion and continued fraction expansion. J. Number Theory 163, 385–405 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fang, L., Wu, M., Li, B.: Beta-expansion and continued fraction expansion of real numbers. Acta Arith. 187(3), 233–253 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Frougny, C., Solomyak, B.: Finite beta-expansions. Ergod. Theory Dyn. Syst. 12(4), 713–723 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gel’fond, A.: A common property of number systems. Izv. Akad. Nauk SSSR. Ser. Mat. 23(6), 809–814 (1959)MathSciNetGoogle Scholar
  19. 19.
    Hill, R., Velani, S.: The ergodic theory of shrinking targets. Invent. Math. 119(1), 175–198 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hofbauer, F.: \(\beta \)-shifts have unique maximal measure. Monatsh. Math. 85(3), 189–198 (1978)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kesseböhmer, M., Stratmann, B.: A multifractal analysis for Stern–Brocot intervals, continued fractions and Diophantine growth rates. J. Reine Angew. Math. 605, 133–163 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kim, D.-H.: The shrinking target property of irrational rotations. Nonlinearity 20(7), 1637–1643 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kong, D.-R., Li, W.-X.: Hausdorff dimension of unique beta expansions. Nonlinearity 28(1), 187–209 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, B., Chen, Y.-C.: Chaotic and topological properties of \(\beta \)-transformations. J. Math. Anal. Appl. 383(2), 585–596 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li, B., Wu, J.: Beta-expansion and continued fraction expansion. J. Math. Anal. Appl. 339(2), 1322–1331 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, B., Persson, T., Wang, B.-W., Wu, J.: Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions. Math. Z. 276(3–4), 799–827 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li, B., Wang, B.-W., Wu, J., Xu, J.: The shrinking target problem in the dynamical system of continued fractions. Proc. Lond. Math. Soc. (3) 108(1), 159–186 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Liao, L.-M., Seuret, S.: Diophantine approximation by orbits of expanding Markov maps. Ergod. Theory Dyn. Syst. 33(2), 585–608 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lü, F., Wu, J.: Diophantine analysis in beta-dynamical systems and Hausdorff dimensions. Adv. Math. 290, 919–937 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Parry, W.: On the \(\beta \)-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 401–416 (1960)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Pfister, C., Sullivan, W.: Large deviations estimates for dynamical systems without the specification property. Applications to the \(\beta \)-shifts. Nonlinearity 18(1), 237–261 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Philipp, W.: Some metrical theorems in number theory. Pac. J. Math. 20, 109–127 (1967)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pollicott, M., Weiss, H.: Multifractal analysis of Lyapunov exponent for continued fraction and Manneville–Pomeau transformations and applications to Diophantine approximation. Commun. Math. Phys. 207(1), 145–171 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8, 477–493 (1957)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Schmeling, J.: Symbolic dynamics for \(\beta \)-shifts and self-normal numbers. Ergod. Theory Dyn. Syst. 17(3), 675–694 (1997)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12(4), 269–278 (1980)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Seuret, S., Wang, B.-W.: Quantitative recurrence properties in conformal iterated function systems. Adv. Math. 280, 472–505 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Tan, B., Wang, B.-W.: Quantitative recurrence properties for beta-dynamical system. Adv. Math. 228(4), 2071–2097 (2011)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Thompson, D.: Irregular sets, the \(\beta \)-transformation and the almost specification property. Trans. Am. Math. Soc. 364(10), 5395–5414 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tong, X., Yu, Y.-L., Zhao, Y.-F.: On the maximal length of consecutive zero digits of \(\beta \)-expansions. Int. J. Number Theory 12(3), 625–633 (2016)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations