Pair correlation of fractional parts derived from rational valued sequences

Abstract

We investigate the pair correlation of the sequence of fractional parts of αxn, n∈N, where xn is rational valued and α is a real number. As examples, we offer two classes of sequences xn whose pair correlation behaves as that of random sequences for almost all real numbers α. First, sequences of the form xn=an/bn where an is lacunary and bn satisfies a certain growth condition. Second, sequences of the form xn=gn/2ω(n) for a positive integer g which is not a power of 2 and where ω(n) denotes the number of distinct prime factors of n.