In this note we study the $\sigma$-pair correlation density \begin{ equation*}R_2^\sigma([a,b], \{ \theta_n \}_n, N)= \frac{ 1}{ N^{ 2-\sigma}} \# \big \{ 1 \leq j \neq k \leq N \, \big| \, \theta_{ j} - \theta_{ k} \in \big [ \frac{ a}{ N^\sigma},\frac{ b}{ N^\sigma} \big ]+ \mathbb Z \big \} \end{ equation*} of a sequence $\{ \theta_n\}_n$ that is equidistributed modulo one for $0 \leq \sigma <2$. The case $\sigma=1$ is commonly referred to as the pair correlation density and the sequence $\{ n^2 \alpha \}_n$ has been of special interest due to its connection to a conjecture of Berry and Tabor on the energy levels of generic completely integrable systems. We prove that if $\alpha$ is Diophantine of type $3-\epsilon$ for every $\epsilon>0$, then for any $0 \leq \sigma <1$ \begin{ align*} \mathrm R_2^\sigma([a,b], \{ \alpha n^2 \}_n, N) \to b-a, \text{ as } N \to \infty. \end{ align*} In this case, we say that the sequence exhibits $\sigma$-pair correlation. In addition to this, we show that for any $0 \leq \sigma < \frac{ 1}{ 4}(9 -\sqrt{ 17})=1.21922...$ there is a set of full Lebesgue measure such that the sequence $\{ \alpha n^2 \}_n$ exhibits $\sigma$-pair correlation.
Categories: math.NT