Pair correlation of fractional parts derived from rational valued sequences

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