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

Abstract

Let $\{a(x){\}}_{x=1}^{\infty }$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\alpha a(x)$ 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 $\{a(x){\}}_{x=1}^{\infty }$ - can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$|n_1(a(x_1)-a(y_1))-n_2(a(x_2)-a(y_2))|<1$$ 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 $N$.

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Cited by in corpus (1)

• Intermediate-scale statistics for real-valued lacunary sequences