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

Abstract

Let $\left\{a\right(x){\}}_{x=1}^{\infty }$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\alpha a\left(x\right)$ 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 $\left\{a\right(x){\}}_{x=1}^{\infty }$—can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$\left|n_1\right(a\left(x_1\right)-a\left(y_1\right))-n_2(a\left(x_2\right)-a\left(y_2\right)\left)\right|<1$$ 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, $$x_1\ne y_1,x_2\ne y_2,(n_1,n_2)\ne (0,0),$$ is at most $N^{4-\delta }$ for some fixed $\delta >0$, for all sufficiently large $N$.