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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

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)