# A SINGULAR FUNCTION FROM STURMIAN CONTINUED FRACTIONS

Kwon, DoYong

• Accepted : 2018.12.06
• Published : 2019.07.01
• 24 1

#### 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)