Pair correlation densities of inhomogeneous quadratic forms, II

Abstract

Denote by $\parallel\cdot\parallel$ the Euclidean norm in $\mathbb {R}\sp k$. We prove that the local pair correlation density of the sequence $\parallel\mathbf {m}-\mathbf {\alpha}\parallel\sp k,\mathbf {m}\in \mathbb {Z}\sp k$, is that of a Poisson process, under Diophantine conditions on the fixed vector $\mathbf {\alpha}\in \mathbb {R}\sp k$ in dimension two, vectors $\mathbf {\alpha}$ of any Diophantine type are admissible; in higher dimensions $(k>2)$, Poisson statistics are observed only for Diophantine vectors of type $\kappa<(k-1)/(k-2)$. Our findings support a conjecture of M. Berry and M. Tabor on the Poisson nature of spectral correlations in quantized integrable systems.