Geometry of Continued Fractions pp 99-114 | Cite as

# Gauss–Kuzmin Statistics

Chapter

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## Abstract

It turns out that the frequency of a positive integer i.e., for a general real

*k*in a continued fraction for almost all real numbers is equal to$$\frac{1}{\ln2}\ln \biggl(1+\frac{1}{k(k+2)} \biggr), $$

*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).## Keywords

Measure Space Continue Fraction Integer Point Multidimensional Case Probability Measure Space
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© Springer-Verlag Berlin Heidelberg 2013