Divisor problems and the pair correlation for the fractional parts of $n^2\alpha$

Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of $n^2\alpha$ is Poissonian for almost all $\alpha$. However, they were not able to find a specific $\alpha$ for which it holds. We show that the problem is related to the problem of determining the number of $(a,b,r)\in {\mathbb{N}}^3$ such that $a\le M$, $b\le N$, $r\le K$ and $p ab \equiv r (q)$for$p$and$q$coprime\mathrm{.}Withsuitableassumptionsontherelativesizeof$K$,$M$,$N$and$q$oneshouldexpecttheretobe$KMN/q$ such triples asymptotically and we will show that this holds on average.