Pair Correlation for Fractional Parts of $αn^2$

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

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