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.