Pair Correlation for Fractional Parts of $αn^2$

D. R. Heath-Brown

Published 2009-04-04Version 1

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