On the value distribution of positive definite quadratic forms

Denote by 0 = lambda(0) < lambda(I) <= lambda(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 - l)-dimensional interval I = I-2 x ... x I-1 we consider the l-level correlation function K-I((I))(R) which counts the number of tuples (i(1),...,i(l)) such that lambda(il,...,lambda)il <= R-2 and lambda(ij) - lambda ij epsilon I-j for 2 <= j <= l. We study the asymptotic behavior of K-I((I))(R) (R) as R tends to infinity. If k >= 4 we prove K-I((I)) (R) similar to c(l)(Q)vol(I)Rlk-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.