Pair correlation densities of inhomogeneous quadratic forms


Under explicit diophantine conditions on $(\alpha,\beta)\in\mathbb{R}^2$, we prove that the local two-point correlations of the sequence given by the values $(m-\alpha)^2+ (n-\beta)^2$, with $(m,n)\in\mathbb{Z}^2$, are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] 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.


Jens Marklof

School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom