二次无理数的连分数的不完全项的统计
Statistics of incomplete quotients of continued fractions of quadratic irrationalities
摘要：
V.I. Arnold用实验证明了二次无理数的部分连分数的不完全项的统计极限与Gauss-Kuz 'min统计极限一致。下面我们简要地证明了这个事实的根方程$r x^2+p x=q$与固定$p$和$r$（$r>0$），并与随机$q$，$q\le R$，$R\to \infty$。在第3节中，我们估计了周期的不完全项之和。根据所得到的界，在到达极限之前，不完全项的平均值在数学上是很小的。我们还估计了“红色”数字在可表示为两个平方和的数字中所占的比例。
英文摘要：
V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact for roots of the equation $r x^2+p x=q$ with fixed $p$ and $r$ ($r>0$), and with random $q$, $q\le R$, $R\to \infty$. 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 average are logarithmically small. We also upper estimate the proportion of the "red" numbers among those representable as a sum of two squares.
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