Jens Marklof
Pair correlation densities of inhomogeneous quadratic forms II
(54K, amslatex)
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.