On the value distribution of positive definite quadratic forms |
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Authors: | Wolfgang Müller |
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Institution: | 1.Institut für Statistik,Technische Universit?t Graz,Graz,Austria |
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Abstract: | Denote by 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l ?1)-dimensional interval I = I 2×. . .×I l we consider the l-level correlation function \({K^{(l)}_I(R)}\) which counts the number of tuples (i 1, . . . , i l ) such that \({\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2}\) and \({\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j}\) for 2 ≤ j ≤ l. We study the asymptotic behavior of \({K^{(l)}_I(R)}\) as R tends to infinity. If k ≥ 4 we prove \({K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}}\) for arbitrary l, where c l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3. |
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