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. |