Approximation by mediants

Author:
Wieb Bosma

Journal:
Math. Comp. **54** (1990), 421-434

MSC:
Primary 11K50; Secondary 11J70

MathSciNet review:
995207

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The distribution is determined of some sequences that measure how well a number is approximated by its mediants (or intermediate continued fraction convergents). The connection with a theorem of Fatou, as well as a new proof of this, is given.

**[1]**W. Bosma, H. Jager, and F. Wiedijk,*Some metrical observations on the approximation by continued fractions*, Nederl. Akad. Wetensch. Indag. Math.**45**(1983), no. 3, 281–299. MR**718069****[2]**P. Erdős,*Some results on diophantine approximation*, Acta Arith.**5**(1959), 359–369 (1959). MR**0121352****[3]**P. Fatou,*Sur l'approximation des incommensurables et les séries trigonométriques*, C. R. Acad. Sci. Paris**139**(1904), 1019-1021.**[4]**Shunji Ito,*Algorithms with mediant convergents and their metrical theory*, Osaka J. Math.**26**(1989), no. 3, 557–578. MR**1021431****[5]**Sh. Ito and H. Nakada,*On natural extensions of transformations related to Diophantine approximations*, Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984), World Sci. Publishing, Singapore, 1985, pp. 185–207. MR**827784****[6]**H. Jager,*The distribution of certain sequences connected with the continued fraction*, Nederl. Akad. Wetensch. Indag. Math.**48**(1986), no. 1, 61–69. MR**834320****[7]**-,*Some metrical observations on the approximation of an irrational number by its nearest mediants*, Preprint.**[8]**J. F. Koksma,*Bewijs van een Stelling over kettingbreuken*, Mathematica A**6**(1937/38), 226-231.**[9]**J. F. Koksma,*On continued fractions*, Simon Stevin**29**(1951/52), 96–102 (1952). MR**0050640****[10]**Hitoshi Nakada, Shunji Ito, and Shigeru Tanaka,*On the invariant measure for the transformations associated with some real continued-fractions*, Keio Engrg. Rep.**30**(1977), no. 13, 159–175. MR**0498461**

Retrieve articles in *Mathematics of Computation*
with MSC:
11K50,
11J70

Retrieve articles in all journals with MSC: 11K50, 11J70

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-0995207-0

Keywords:
Continued fractions,
Diophantine approximation

Article copyright:
© Copyright 1990
American Mathematical Society