DOI QR코드

DOI QR Code

A SINGULAR FUNCTION FROM STURMIAN CONTINUED FRACTIONS

Kwon, DoYong

  • Received : 2018.08.25
  • Accepted : 2018.12.06
  • Published : 2019.07.01

Abstract

For ${\alpha}{\geq}1$, let $s_{\alpha}(n)={\lceil}{\alpha}n{\rceil}-{\lceil}{\alpha}(n-1){\rceil}$. A continued fraction $C({\alpha})=[0;s_{\alpha}(1),s_{\alpha}(2),{\ldots}]$ is considered and analyzed. Appealing to Diophantine approximation, we investigate the differentiability of $C({\alpha})$, and then show its singularity.

Keywords

singular function;continued fraction;Diophantine approximation;Sturmian word

References

  1. J.-P. Allouche, J. L. Davison, M. Queffelec, and L. Q. Zamboni, Continued fractions, J. Number Theory 91 (2001), no. 1, 39-66. https://doi.org/10.1006/jnth.2001.2669
  2. J.-P. Borel and F. Laubie, Quelques mots sur la droite projective reelle, J. Theor. Nombres Bordeaux 5 (1993), no. 1, 23-51.
  3. G. Cantor, De la puissance des ensembles parfaits de points, Acta Math. 4 (1884), no. 1, 381-392. https://doi.org/10.1007/BF02418423
  4. A. Denjoy, Sur une fonction reelle de Minkowski, J. Math. Pures Appl. 17 (1938) 105-151.
  5. A. A. Dushistova, I. D. Kan, and N. G. Moshchevitin, Differentiability of the Minkowski question mark function, J. Math. Anal. Appl. 401 (2013), no. 2, 774-794. https://doi.org/10.1016/j.jmaa.2012.12.058
  6. V. Jarnik, Zur metrischen Theorie der diophantischen Approximationen, Prace mat.-fiz. 36 (1929), 91-106.
  7. A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, IL, 1964.
  8. D.Y. Kwon, A devil's staircase from rotations and irrationality measures for Liouville numbers, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 739-756. https://doi.org/10.1017/S0305004108001606
  9. D.Y. Kwon, Moments of discrete measures with dense jumps induced by ${\beta}$-expansions, J. Math. Anal. Appl. 399 (2013), no. 1, 1-11. https://doi.org/10.1016/j.jmaa.2012.07.014
  10. D.Y. Kwon, A one-parameter family of Dirichlet series whose coefficients are Sturmian words, J. Number Theory 147 (2015), 824-835.
  11. D.Y. Kwon, The fractional totient function and Sturmian Dirichlet series, Honam Math. J. 39 (2017), no. 2, 297-305. https://doi.org/10.5831/HMJ.2017.39.2.297
  12. M. Lothaire, Algebraic combinatorics on words, Cambridge University Press, Cambridge, 2002.
  13. J. Paradis, P. Viader, and L. Bibiloni, The derivative of Minkowski's ?(x) function, J. Math. Anal. Appl. 253 (2001), no. 1, 107-125. https://doi.org/10.1006/jmaa.2000.7064
  14. W. Parry, On the ${\beta}$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.
  15. A. M. Rockett and P. Szusz, Continued Fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.
  16. K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20; corrigendum, 168.
  17. R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc. 53 (1943), 427-439.
  18. J. Sondow, An irrationality measure for Liouville numbers and conditional measures for Euler's constant, 23rd Journees Arithmetiques, Graz, Austria, 2003. Available at http://arxiv.org/abs/math/0307308.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)