The metric theory of the pair correlation function of real-valued lacunary sequences

Let {a(x)}(x=1)(infinity) be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations alpha a(x) is Poissonian for Lebesgue almost every alpha is an element of R. By using harmonic analysis, our result-irrespective of the choice of the real-valued sequence {a(x)}(x=1)(infinity)-can essentially be reduced to showing that the number of solutions to the Diophantine inequality vertical bar n(1)(a(x(1)) - a(y(1))) - n(2)(a(x(2)) - a(y(2)))vertical bar < 1 in integer six-tuples (n(1), n(2), x(1), x(2), y(1), y(2)) located in the box [-N, N](6) with the "excluded diagonals"; that is, x(1) not equal y(1), x(2) not equal y(2), (n(1), n(2)) not equal (0,0), is at most N4-8 for some fixed delta > 0, for all sufficiently large N.