On the $σ$-Pair Correlation Density of Quadratic Sequences Modulo One
In this note we study the $σ$-pair correlation density \begin{equation*}R_2^σ([a,b], \{ θ_n \}_n, N)= \frac{1}{N^{2-σ}} \# \big \{ 1 \leq j \neq k \leq N \, \big| \, θ_{j} - θ_{k} \in \big [ \frac{a}{N^σ},\frac{b}{N^σ} \big ]+ \mathbb Z \big \} \end{equation*} of a sequence $\{ θ_n\}_n$ that is equidistributed modulo one for $0 \leq σ<2$. The case $σ=1$ is commonly referred to as the pair correlation density and the sequence $\{ n^2 α\}_n$ 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 $α$ is Diophantine of type $3-ε$ for every $ε>0$, then for any $0 \leq σ<1$ \begin{align*} \mathrm R_2^σ([a,b], \{ αn^2 \}_n, N) \to b-a, \text{ as } N \to \infty. \end{align*} In this case, we say that the sequence exhibits $σ$-pair correlation. In addition to this, we show that for any $0 \leq σ< \frac{1}{4}(9 -\sqrt{17})=1.21922...$ there is a set of full Lebesgue measure such that the sequence $\{ αn^2 \}_n$ exhibits $σ$-pair correlation.