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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.
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참고문헌 (18)
참고문헌 신청
J. -P. Allouche / 2001 / Continued fractions / J. Number Theory 91(1):39~66
J.-P. Borel / 1993 / Quelques mots sur la droite projective r´eelle / J. Theor. Nombres Bordeaux 5(1):23~51
G. Cantor / 1884 / De la puissance des ensembles parfaits de points / Acta Math 4(1):381~392
A. Denjoy / 1938 / Sur une fonction r´eelle de Minkowski / J. Math. Pures Appl 17:105~151
A. A. Dushistova / 2013 / Differentiability of the Minkowski question mark function / J. Math. Anal. Appl 401(2):774~794
J.-P. Borel / 1993 / Quelques mots sur la droite projective r´eelle / J. Theor. Nombres Bordeaux 5(1):23~51
G. Cantor / 1884 / De la puissance des ensembles parfaits de points / Acta Math 4(1):381~392
A. Denjoy / 1938 / Sur une fonction r´eelle de Minkowski / J. Math. Pures Appl 17:105~151
A. A. Dushistova / 2013 / Differentiability of the Minkowski question mark function / J. Math. Anal. Appl 401(2):774~794
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