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.