Pair correlation densities of inhomogeneous quadratic forms II

Abstract

Denote by $\| \cdot \|$ the euclidean norm in \RR^k. We prove that the local pair correlation density of the sequence \| \vecm -\vecalf \|^k, \vecm\in\ZZ^k, is that of a Poisson process, under diophantine conditions on the fixed vector \vecalf\in\RR^k: in dimension two, vectors \vecalf of any diophantine type are admissible; in higher dimensions ($k>2$), Poisson statistics are only observed for diophantine vectors of type $\kappa<(k-1)/(k-2)$. Our findings support a conjecture of Berry and Tabor on the Poisson nature of spectral correlations in quantized integrable systems