模1序列的长程相关
Long-Range Correlations of Sequences Modulo 1
摘要：
在本文中，我们考虑一般序列的小数部分，例如序列$α\sqrt{n}$或$αn^2$。我们给出了一个一般的方法，它允许一个显示，长程相关性（相关的支持检验函数增长，因为我们认为更多的点）是泊松。我们表明，这些声明的收敛性可以减少到相关的外尔和的界限。特别是，我们将这种方法应用于上述示例。在这样做的过程中，我们恢复了Technau-Walker（2020）关于$αn^2$的三重相关的最新结果，并将结果推广到更高的矩。对于上述两个序列，这是指示更高级（$m \ge 3$）相关性的伪随机性质的仅有结果之一。
英文摘要：
In this paper we consider the fractional parts of a general sequence, for example the sequence $α\sqrt{n}$ or $αn^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $αn^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations.
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