Geodesic
Continued fractions of square roots of~rational numbers and their statistics
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 5, pp. 843-855

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The differences are described between the statistics of the periodic continued fractions of quadratic irrational numbers and the universal Gauss–Kuzmin statistics of the continued fractions of random real numbers. Besides being a survey of these differences, this article discusses the results of V. A. Bykovskii [Bykovsky] and his school, who have proved the author's 1993 conjectures on the similarities of these statistics. 
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     author = {V. I. Arnol'd},
     title = {Continued fractions of square roots of~rational numbers and their statistics},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {843--855},
     publisher = {mathdoc},
     volume = {62},
     number = {5},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2007_62_5_a0/}
}
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V. I. Arnol'd. Continued fractions of square roots of~rational numbers and their statistics. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 5, pp. 843-855. http://geodesic.mathdoc.fr/item/RM_2007_62_5_a0/