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8 ANDERS SÖDERGRENwhere j B = minj∈B j.Remark 4. The formula in (10) can also be obtained from the mixed cumulants ofN(V 1 ), . . . , N(V k ) (see e.g. [5, Prop. 6.16]).We are interested in comparing the formula in (10) with the result of Theorem 3.We first need the following observation.Lemma 3. Let D(k) be the set of matrices D which have nonnegative entries andare on the form occurring in Theorem 3, together with the k × k indentity matrix I k .When D = I k let ν i = i, 1 ≤ i ≤ k. Then there is a bijection g : D(k) → P(k) withthe property that if D ∈ D(k) is an m × k matrix and g(D) = P = {B 1 , . . . , B #P }then #P = m and {ν 1 , . . . , ν m } = {j B1 , . . . , j Bm }.Proof. For D ∈ D(k) of size m × k we set B i = {j | d ij ≠ 0}, 1 ≤ i ≤ m,and define g(D) = {B 1 , . . . , B m }. It is clear that g is a bijection with the desiredproperties.□This lemma together with Theorem 3 and the result in (10) immediately impliesthe following theorem.Theorem 4. Let Ñj(L) = 1 2 N j(L) so that Ñj(L) denotes the number of pairs of nonzerolattice points of L belonging to the n-ball centered at the origin having volumeV j , 1 ≤ j ≤ k. Thenas n → ∞.( k∏Ej=1)Ñ j (L) → E( k∏j=1)N(V j )Corollary 1. Let k ≥ 1 and let V = (V 1 , V 2 , . . . , V k ) for some fixed 0 < V 1