This article is cited in 2 scientific papers (total in 3 papers)
Statistics of the periods of continued fractions for quadratic irrationals
V. I. Arnol'd
Steklov Mathematical Institute, Russian Academy of Sciences
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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Izvestiya: Mathematics, 2008, 72:1, 1–34
MSC: 11A55, 37A45
V. I. Arnol'd, “Statistics of the periods of continued fractions for quadratic irrationals”, Izv. RAN. Ser. Mat., 72:1 (2008), 3–38; Izv. Math., 72:1 (2008), 1–34
Citation in format AMSBIB
\paper Statistics of the periods of continued fractions for quadratic irrationals
\jour Izv. RAN. Ser. Mat.
\jour Izv. Math.
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This publication is cited in the following articles:
“Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030
Arnold V.I., “Lengths of periods of continued fractions of square roots of integers”, Funct. Anal. Other Math., 2:2-4 (2009), 151–164
Iavernaro F., Trigiante D., “Continued fractions without fractions: Lagrange theorem and Pell equations”, Nonlinear Analysis-Theory Methods & Applications, 71:12 (2009), E2136–E2151
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