Pair Correlation for Fractional Parts of $\alpha n^2$

TL;DR

This paper constructs specific real numbers with quadratic fractional parts exhibiting Poissonian pair correlation and analyzes the behavior for Diophantine numbers, providing bounds on the correlation function.

Contribution

It introduces a method to construct real numbers with quadratic fractional parts that have Poissonian pair correlation and establishes bounds for Diophantine numbers.

Findings

Constructed real numbers with pair correlation tending to X

Proved bounds on pair correlation for Diophantine numbers

Demonstrated Poissonian behavior in fractional quadratic sequences

Abstract

We construct real numbers $\alpha$ for which the pair correlation function \[N^{-1}#\{m<n\le N:||\alpha m^2-\alpha n^2||\le XN^{-1}\}\] tends to $X$ as $N$ grows. Moreover we show for any "Diophantine" $\alpha$ that the pair correlation function is $X+O(X^{7/8})+O((\log N)^{-1}$ for $1\le X\le\log N$.