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

Abstract

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 $pab\equiv r(q)$ for $p$ and $q$ coprime. With suitable assumptions on the relative size of $K$, $M$, $N$ and $q$ one should expect there to be $KMN/q$ such triples asymptotically and we will show that this holds on average.

27 pages