Statistics of the periods of continued fractions for quadratic irrationals

V. I. Arnol'd

Izvestiya. Mathematics , Tome 72 (2008) no. 1, pp. 1-34.

Résumé

The distribution of frequencies of elements of continued fractions for random real numbers was obtained by Kuz'min in 1928 and is therefore referred to as Gauss–Kuz'min statistics. An old conjecture of the author states that the elements of periodic continued fractions of quadratic irrationals satisfy the same statistics in the mean. This was recently proved by Bykovsky and his students. In this paper we complement those results by a study of the statistics of the period lengths of continued fractions for quadratic irrationals. In particular, this theory implies that the elements forming the periods of continued fractions of the roots $x$ of the equations $x^2+px+q=0$ with integer coefficients do not exhaust the set of all random sequences whose elements satisfy the Gauss–Kuz'min statistics. For example, these sequences are palindromic, that is, they read the same backwards as forwards.

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