论文标题
序列模拟1的远程相关性1
Long-Range Correlations of Sequences Modulo 1
论文作者
论文摘要
在本文中，我们考虑了一般序列的分数部分，例如序列$α\ sqrt {n} $或$αn^2 $。我们给出了一种一般方法，该方法允许人们表明远程相关性（随着我们考虑更多点的增长的支持增长的相关性）是Poissonian。我们表明，这些有关收敛性的陈述可以减少到相关的Weyl和相关的界限。特别是我们将此方法应用于上述示例。这样一来，我们收回了Technau-Walker（2020）的最新结果，即$αn^2 $的三重相关性，并将结果推广到更高的矩。对于上述两个序列，这是表明较高级别（$ m \ ge 3 $）相关性的伪随机性质的唯一结果之一。
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.
