  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 $1\le X\le\log N$.
