Jens Marklof
Pair correlation densities of inhomogeneous quadratic forms
(119K, amslatex)
ABSTRACT. Under explicit diophantine conditions on $(\alpha,\beta)\in\RR^2$, we
prove that the local two-point correlations of the
sequence given by the values $(m-\alpha)^2+(n-\beta)^2$, with
$(m,n)\in\ZZ^2$, are those of a Poisson process.
This partly confirms a conjecture
of Berry and Tabor on spectral statistics of quantized
integrable systems, and also establishes a particular case of the
quantitative version of the Oppenheim conjecture for
inhomogeneous quadratic forms of signature (2,2).
The proof uses theta sums and Ratner's classification of measures
invariant under unipotent flows.