I did not look at the paper, but the lim_{|K| -> 0} k_i k_j = lim_{|K| -> 0} K_i K_j / |K|^2 does not seem to exist.

For i=1 and j=2 the subsequence K_1=K_2=K_3 =1/n yields a different limit (which is 1/3) than the subsequence K_1=K_2=0 and K_3=1/n (which is 0)

I agree that something seems strange about the limit in this form. In the setting where it is used, K is a reciprocal lattice vector in Fourier space. The limit is used for determining the contribution of the Fourier transform of a function for K=0. Possibly that allows the limit to be viewed differently.

Robert, that could very well be the case. We are looking at the transform of a function which is periodic with respect to a box L, so K_i = 2*pi*n/L_i, where n is an integer, and taking K=0 in the transform represents the mean level of the function.

To me it is not entirely clear how such a vector goes to zero, perhaps assuming some uniform distribution is valid. How do you that the average of the limit is delta_{ij}/3 in that case?

continuing with the idea of Robert, if we express the vector k in an orthonormal basis {e1, e2, e3}, then we have:

k = (k1,k2,k3) = Ik1I*e1+Ik2I*e2+Ik3I*e3

where the scalar product k·k= Ik1I^2+Ik2I^2+Ik3I^2 = 1

Then the scalar product

ki · kj = delta (i, j) * IkiI * IkjI

Now, averaging, we will have that Ik1I = Ik2I = Ik3I, and therefore

Ik1I^2+Ik2I^2+Ik3I^2 = 3*IkiI^2 = 1

IkiI ^ 2 = 1/3

After having been in touch with the authors of the paper I can confirm that what Robert has already suggested is the case here. The tensor k_i k_j is considered to be isotropic in the limit, which gives the result I was asking about. The general case is not true, as has also been pointed out.