  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.
