Pair correlation of fractional parts derived from rational valued sequences, II

Abstract

A rational valued vector sequence x→, for some fixed k and r∈N, is a map x→:Nk→Qr. In the present paper, we complement the results of [3] with a discussion on rational valued vector sequences. We investigate the pair correlation function for the fractional parts of sequences t→⋅x→, where x→ is a rational valued vector sequence and t→∈Rr. We offer a new class of sequences x→ whose pair correlation function behaves as that of random sequences for almost all real vectors t→, namely, injective sequences of the formx→(m,n)=(p1mp2nb→(m,n),p3mp4nb→(m,n)) for p1, p2, p3, and p4 primes and where b→:N2→N satisfies a certain growth condition.