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A metrical result on the approximation by continued fractions


Author: H. Jager
Journal: Math. Comp. 81 (2012), 2377-2382
MSC (2010): Primary 11K50
DOI: https://doi.org/10.1090/S0025-5718-2012-02621-4
Published electronically: May 22, 2012
MathSciNet review: 2945161
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Abstract: Let $ x$ be a real irrational number and $ p_n/q_n, \,\,n=0,1,2,\ldots $ its sequence of continued fraction convergents. Define $ \theta _n=q_n\big \vert q_nx-p_n\big \vert.$ For almost all $ x$ the distribution function of the sequence $ \big \vert\theta _{n+1}-\theta _{n-1}\big \vert, \,\,n=0,1,2,\ldots $ is determined and its expectation calculated.


References [Enhancements On Off] (What's this?)

  • 1. Karma Dajani and Cor Kraaikamp, Ergodic theory of numbers, Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, 2002. MR 1917322
  • 2. J. F. Koksma, Diophantische Approximationen, Springer-Verlag, Berlin-New York, 1974 (German). Reprint. MR 0344200
  • 3. 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

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Additional Information

H. Jager
Affiliation: Oude Larenseweg 26, 7214 PC Epse, the Netherlands
Email: epserbos@xs4all.nl

DOI: https://doi.org/10.1090/S0025-5718-2012-02621-4
Keywords: Continued fractions, approximation coefficients, metrical theory
Received by editor(s): May 19, 2011
Published electronically: May 22, 2012
Article copyright: © Copyright 2012 American Mathematical Society