论文标题
代数数字是否遵循Khinchin定律？
Do algebraic numbers follow Khinchin's Law?
论文作者
论文摘要
随机数的常规持续分数的系数是根据Khinchin定律通过高斯-Kuzmin分布分布的。它们的几何平均值会收敛到Khinchin的常数，而其合理近似速度是Khinchin的速度。这些定理是否也适用于代数数量$> 2 $是一个空旷的问题。但是，由于它们几乎适用于所有数字，因此通常可以推断出，非二次代数数字也很可能也这样做。我们认为这种推论不是很好的扎根。有很大的数值证据表明Khinchin的速度太快了。对于Khinchin的定律和Khinchin的常数，数值证据尚不清楚。我们应用Kullback Leibler Divergence（KLD）表明高斯-Kuzmin分布不太适合$> 2 $的代数数量。我们的建议将有限零件的高斯 - 库兹敏分布截断略微略有拟合，但其KLD仍然比随机数的KLD大得多。因此，如果收敛性的收敛性是不统一的，并且每个代数号都有其自身的界限。我们得出的结论是，没有证据将持续数字的定理应用于代数数。
The coefficients of the regular continued fraction for random numbers are distributed by the Gauss-Kuzmin distribution according to Khinchin's law. Their geometric mean converges to Khinchin's constant and their rational approximation speed is Khinchin's speed. It is an open question whether these theorems also apply to algebraic numbers of degree $>2$. Since they apply to almost all numbers it is, however, commonly inferred that it is most likely that non quadratic algebraic numbers also do so. We argue that this inference is not well grounded. There is strong numerical evidence that Khinchin's speed is too fast. For Khinchin's law and Khinchin's constant the numerical evidence is unclear. We apply the Kullback Leibler Divergence (KLD) to show that the Gauss-Kuzmin distribution does not fit well for algebraic numbers of degree $>2$. Our suggestion to truncate the Gauss-Kuzmin distribution for finite parts fits slightly better but its KLD is still much larger than the KLD of a random number. So, if it converges the convergence is non uniform and each algebraic number has its own bound. We conclude that there is no evidence to apply the theorems that hold for random numbers to algebraic numbers.
