Pair correlation densities of inhomogeneous quadratic forms

Details Pair correlation densities of inhomogeneous quadratic forms Pair correlation densities of inhomogeneous quadratic forms

2002-10-14

arxiv.org/abs/math/0210197v2

Ann. of Math. (2), Vol. 158 (2003), no. 2, 419--471

Mathematics

Number Theory

53 pages published version

Scientific paper

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 $(m-\alpha)^2+\break (n-\beta)^2$, with $(m,n)\in\ZZ^2$, 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.

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