Pair correlation of fractional parts derived from rational valued sequences

We investigate the pair correlation of the sequence of fractional parts of , , where is rational valued and is a real number. As examples, we offer two classes of sequences whose pair correlation behaves as that of random sequences for almost all real numbers . First, sequences of the form where is lacunary and satisfies a certain growth condition. Second, sequences of the form for a positive integer which is not a power of 2 and where denotes the number of distinct prime factors of .

我们研究\(\{n\alpha\}\)序列的对关联，其中\(\alpha\)是有理数值，\(n\)是一个实数。作为例子，我们给出两类序列\(\{x_n\}\)，对于几乎所有的实数\(\alpha\)，它们的对关联表现得如同随机序列的对关联。首先，形如\(x_n = \lambda^n\alpha\)的序列，其中\(\{\lambda^n\}\)是缺项的且\(\lambda\)满足某个增长条件。其次，形如\(x_n = n\alpha\)的序列，其中\(n\)是一个正整数且不是\(2\)的幂，并且\(\omega(n)\)表示\(n\)的不同素因子的个数。