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

TL;DR

This paper explores the pair correlation of fractional parts of quadratic sequences, linking it to divisor problems, and demonstrates average-case asymptotic behavior under certain conditions.

Contribution

It establishes a connection between pair correlation problems and divisor counting, providing average-case results for specific parameter ranges.

Findings

Pair correlation for fractional parts of n^2 alpha is Poissonian for almost all alpha.

The problem reduces to counting solutions of a divisor-related congruence.

On average, the number of solutions matches the expected asymptotic formula.

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 $(a,b,r) \in \N^3$ such that $a \le M$, $b \le N$, $r \le K$ and $p ab \equiv r (q)$ 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.