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

Let $\{a(x){\}}_{x=1}^{\mathrm{\infty }}$ 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 $\alpha \in \mathbb{R}$. By using harmonic analysis, our result - irrespective of the choice of the real-valued sequence $\{a(x){\}}_{x=1}^{\mathrm{\infty }}$ - can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$\mathrm{\mid }{n}_1(a(x_1)-a(y_1))-n_2(a(x_2)-a(y_2))\mathrm{\mid }<1$$ 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 $$x_1\mathrm{\ne }{y}_1,x_2\mathrm{\ne }{y}_2,(n_1,n_2)\mathrm{\ne }(0,0),$$ is at most $N^{4-\delta }$ for some fixed $\delta >0$, for all sufficiently large $N$.