논문KCI 등재
A singular function from Sturmian continued fractions
대한수학회 (The Korean Mathematical Society)|2019.07|대한수학회지 (Journal of the KMS)|제56권 제4호|pp. 1049-1061|DOI 10.4134/JKMS.j180578
2
조회
0
다운로드
1
인용
0
참고문헌
키워드
주제분류수학
초록
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.
Abstract
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.
참고문헌 (18)
- J. -P. Allouche, "Continued fractions", J. Number Theory, 91(1), pp.39~66 (2001)
- J.-P. Borel, "Quelques mots sur la droite projective r´eelle", J. Theor. Nombres Bordeaux, 5(1), pp.23~51 (1993)
- G. Cantor, "De la puissance des ensembles parfaits de points", Acta Math, 4(1), pp.381~392 (1884)
- A. Denjoy, "Sur une fonction r´eelle de Minkowski", J. Math. Pures Appl, 17, pp.105~151 (1938)
- A. A. Dushistova, "Differentiability of the Minkowski question mark function", J. Math. Anal. Appl, 401(2), pp.774~794 (2013)
- V. Jarn´ık, "Zur metrischen Theorie der diophantischen Approximationen", Prace mat. -fiz, 36, pp.91~106 (1929)
- A. Ya. Khinchin, "Continued Fractions" (1964), The University of Chicago Press
- D.Y. Kwon, "A devil’s staircase from rotations and irrationality measures for Liouville numbers", Math. Proc. Cambridge Philos. Soc, 145(3), pp.739~756 (2008)
- D.Y. Kwo, "Moments of discrete measures with dense jumps induced by β-expansions", J. Math. Anal. Appl, 399(1), pp.1~11 (2013)
- D.Y. Kwo, "A one-parameter family of Dirichlet series whose coefficients are Sturmian words", J. Number Theory, 147, pp.824~835 (2015)
- 권도용, "The fractional totient function and Sturmian Dirichlet series", 호남수학학술지, 39(2), pp.297~305 (2017), 호남수학회
- M. Lothaire, "Algebraic combinatorics on words" (2002), Cambridge University Press
- J. Parad´ıs, "The derivative of Minkowski’s ?(x)function", J. Math. Anal. Appl, 253(1), pp.107~125 (2001)
- W. Parry, "On the β-expansions of real numbers", Acta Math. Acad. Sci. Hungar, 11, pp.401~416 (1960)
- A. M. Rockett, "Continued Fractions" (1992), World Scientific Publishing Co., Inc
- K. F. Roth, "Rational approximations to algebraic numbers", Mathematika, 2, pp.1~20 (1955)[DOI]
- R. Salem, "On some singular monotonic functions which are strictly increasing", Trans. Amer. Math. Soc, 53, pp.427~439 (1943)
- "An irrationality measure for Liouville numbers and conditional measures for Euler’s constant" (2003)[URL]
관련 논문AI추천
Differentiability and non-differentiability points of the Minkowski question mark functionIN-SOO BAEK대한수학회 · 대한수학회논문집2016 · 제31권 제4호DISCRETE MEASURES WITH DENSE JUMPS INDUCED BY STURMIAN DIRICHLET SERIES권도용대한수학회 · 대한수학회보2015 · 제52권 제6호Continued fractions and the density of graphs of some functions이정연, 전병흡, 채희준강원경기수학회 · 한국수학논문집2017 · 제25권 제2호A note on the growth rate of Engel digits대한수학회 · 대한수학회논문집2026 · 제41권 제1호Distribution of the approximation exponents of a family of power series over a finite fieldIheb Elouaer, Khalil Ayadi대한수학회 · 대한수학회보2018 · 제55권 제6호On real continued fractionsA. Dalloul장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2020 · 제23권 제1호
이 간행물의 다른 논문 (13건)
On a class of quasilinear elliptic equation with indefinite weights on graphsGuoqing Zhang, Shoudong Manpp. 857-867A characterization of class groups via sets of lengthsAlfred Geroldinger, Wolfgang Alexander Schmidpp. 869-915Homogeneity and symmetry on almost Kenmotsu $3$-manifoldsYaning Wangpp. 917-934Limit theorems for Hawkes processes with uniform immigrants설영수pp. 935-946Shifted tableau switchings and shifted Littlewood-Richardson coefficients최승일, 오영탁, 남선영pp. 947-984