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

Abstract

We construct real numbers $\alpha$ 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" $\alpha$ that the pair correlation function is $X+O(X^{7/8})+O((\log N{)}^{-1}$ for $1\le X\le \log N$.

References in corpus (1)

• The distribution of spacings between the fractional parts of $n^2\alpha$

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