Gauss–Kuzmin Statistics

  • Oleg Karpenkov
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)


It turns out that the frequency of a positive integer k in a continued fraction for almost all real numbers is equal to
$$\frac{1}{\ln2}\ln \biggl(1+\frac{1}{k(k+2)} \biggr), $$
i.e., for a general real x we have 42 % of 1, 17 % of 2, 9 % of 3, etc. This distribution is traditionally called the Gauss–Kuzmin distribution. In this chapter we describe two strategies to study distributions of elements in continued fractions. A classical approach to the Gauss–Kuzmin distribution is based on the ergodicity of the Gauss map. The second approach is related to the geometry of continued fractions and its projective invariance. It is interesting to note that the frequencies of elements has an unexpected interpretation in terms of cross-ratios. Further we generalize the second approach to the multidimensional case (see Chap.  19).


Measure Space Continue Fraction Integer Point Multidimensional Case Probability Measure Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oleg Karpenkov
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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