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J. Korean Math. Soc. 2019; 56(4): 1049-1061

Online first article February 12, 2019      Printed July 1, 2019

https://doi.org/10.4134/JKMS.j180578

Copyright © The Korean Mathematical Society.

A singular function from Sturmian continued fractions

DoYong Kwon

Chonnam National University

Abstract

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For $\alpha\geq1$, 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

MSC numbers: 26A30, 11A55, 11J04, 68R15

Supported by: This research was supported by Basic Science Research Program through the National Resear ch Foundation of Korea (NRF) funded by the Ministry of Education (grant number: NRF-2016R1D1A1B020 10219).

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