Let $x$ be a quadratic irrational and let $\mathbb{P}$ be the set of prime numbers. We show the existence of an infinite set $S\subset \mathbb{P}$ such that the statistics of the period of the continued fraction expansions along the sequence $\left\{px:p\in S\right\}$ approach the ‘normal’ statistics given by the Gauss–Kuzmin measure. Under the generalized Riemann hypothesis, we prove that there exist full density subsets $S\subset \mathbb{P}$ and $T\subset \mathbb{N}$ satisfying the same assertion. We give a rate of convergence in all cases.

中文翻译：

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二次算术序列的连续分数

让 $X$ 成为二次非理性的，让 $\mathbb{P}$是素数的集合。我们证明了无限集的存在$\mathrm{小号}\subset \mathbb{P}$ 这样连续分数扩展的周期的统计就沿着序列 $\left\{pX：p\in \mathrm{小号}\right\}$接近高斯-库兹敏测度给出的“正常”统计量。在广义黎曼假设下，我们证明存在完全密度子集$\mathrm{小号}\subset \mathbb{P}$ 和 $Ť\subset \mathbb{ñ}$满足相同的主张。我们给出所有情况下的收敛速度。