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Mathematics of Computation

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Approximation by mediants

Author: Wieb Bosma
Journal: Math. Comp. 54 (1990), 421-434
MSC: Primary 11K50; Secondary 11J70
MathSciNet review: 995207
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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.

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  • [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

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Keywords: Continued fractions, Diophantine approximation
Article copyright: © Copyright 1990 American Mathematical Society

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