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- W1856773822 abstract "AbstractV.I. Arnold has experimentally established that the limit of the statistics of incomplete quotientsof partial continued fractions of quadratic irrationalities coincides with the Gauss–Kuz’min statistics.Below we briefly prove this fact for roots of the equation rx 2 + px = q with fixed p and r (r > 0), andwith random q, q ≤ R, R → ∞. In Section 3 we estimate the sum of incomplete quotients of the period.According to the obtained bound, prior to the passage to the limit, incomplete quotients in averageare logarithmically small. We also upper estimate the proportion of the “red” numbers among thoserepresentable as a sum of two squares. Keywords: periodic continued fractions, Arnold’s conjecture, Gauss–Kuz’min statistics, Bykovskii’s theo-rem, Farey fraction, Pell’s equation.MSC classes: 11T06; 11T24; 37E15. 1 The statement of the main results and their consideration The papers, monographs, and reports of V.I. Arnold dedicated to the statistics of periods of continuedfractions and their multidimensional generalizations (see [1]–[7]) contain a vast number of experimental factsand outline prospects for further investigations. The first step in this direction is the proof of the followingassertion (established by V.I. Arnold): the statistics of incomplete quotients of the continued fraction ofsolutions to a random quadratic equation with integer coefficients, proceeding to the “thermodynamic” limit,turns into the Gauss–Kuz’min statistics. Unfortunately, due to circumstances the proof of this assertionperformed by V.A. Bykovskii and his followers was not published. Below we adduce an elementary proof ofthe one-parameter version of this assertion.Any number x ∈ R is representable as a continuous fraction in the formx = a" @default.
- W1856773822 created "2016-06-24" @default.
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- W1856773822 date "2008-10-03" @default.
- W1856773822 modified "2023-09-27" @default.
- W1856773822 title "Statistics of incomplete quotients of continued fractions of quadratic irrationalities" @default.
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