Pair Correlation Densities of Inhomogeneous Quadratic Forms
نویسندگان
چکیده
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.
منابع مشابه
Pair Correlation Densities of Inhomogeneous Quadratic Forms Ii
Abstract. Denote by ‖ · ‖ the euclidean norm in R. We prove that the local pair correlation density of the sequence ‖m− α‖k, m ∈ Z, is that of a Poisson process, under diophantine conditions on the fixed vector α ∈ R: in dimension two, vectors α of any diophantine type are admissible; in higher dimensions (k > 2), Poisson statistics are only observed for diophantine vectors of type κ < (k − 1)/...
متن کاملCorrection to “pair Correlation Densities of Inhomogeneous Quadratic Forms, Ii”
Proof One follows the argument in [1, pp. 432 – 433] (cf. also [2, Sec. 9]) to show that for each fixed badly approximable (k − 2)-tuple (α1, . . . , αk−2) there is a set of second Baire category of (αk−1, αk) ∈ T2 such that conditions (i), (ii), and (iii) of Theorem 1.7 hold for α = (α1, . . . , αk). Because the set of badly approximable (k − 2)-tuples is dense in Tk−2, and the set of second B...
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