TAILIEUCHUNG - Đề tài " Pair correlation densities of inhomogeneous quadratic forms "

Under explicit diophantine conditions on (α, β) ∈ R2 , we prove that the local two-point correlations of the sequence given by the values (m − α)2 + (n−β)2 , with (m, n) ∈ Z2 , 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. . | Annals of Mathematics Pair correlation densities of inhomogeneous quadratic forms By Jens Marklof Annals of Mathematics 158 2003 419-471 Pair correlation densities of inhomogeneous quadratic forms By Jens Marklof Abstract Under explicit diophantine conditions on a ft G R2 we prove that the local two-point correlations of the sequence given by the values m a 2 n ft 2 with m n gZ2 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. 1. Introduction . Let us denote by 0 X1 X2 x the infinite sequence given by the values of m a 2 n ft 2 at lattice points m n G z2 for fixed a ft G 0 1 . In a numerical experiment Cheng and Lebowitz 3 found that for generic a ft the local statistical measures of the deterministic sequence Xj appear to be those of independent random variables from a Poisson process. . This numerical observation supports a conjecture of Berry and Tabor 2 in the context of quantum chaos according to which the local eigenvalue statistics of generic quantized integrable systems are Poissonian. In the case discussed here the Xj may be viewed up to a factor 4n2 as the eigenvalues of the Laplacian _ _ d 2 cft_ dx2 dy2 with quasi-periodicity conditions ifftx k y l e 2ni ak mv x y k l a. The corresponding classical dynamical system is the geodesic flow on the unit tangent bundle of the flat torus T2. 420 JENS MARKLOF . The asymptotic density of the sequence of Xj is n according to the well known formula for the number of lattice points in a large shifted circle j Xj A m n 2 m a 2 n 3 2 X nX for X ữQ. The rate of convergence is discussed in detail by Kendall 11 . . More generally suppose we have a sequence Xi X2 w of mean .

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