| 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){K^{(l)}_I(R)} which counts the number of tuples (i
1, . . . , i
l
) such that li1,?,lil ￡ R2{\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2} and lij-li1 ? Ij{\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j} for 2 ≤ j ≤ l. We study the asymptotic behavior of K(l)I(R){K^{(l)}_I(R)} as R tends to infinity. If k ≥ 4 we prove K(l)I(R) ~ cl(Q) vol(I)Rlk-2(l-1){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. |
