[math-fun] Intersection of random cosets of Z[exp(2πi/3)]: When is this set empty?

Let E denote the triangular lattice Z[w] of Eisenstein integers, where w = -1/2 + i sqrt(3/4) is a primitive 1^(1/3) in C. For j = 1, 2, ...,n let A_j = K_j + L_j w ≠ 0 and B_j = M_j + N_j w be arbitrary elements of E with all A_j nonzero. Multiplication by A_j carries E into a subset of itself by uniform scaling, and translation by B_j carries E onto itself isometrically. Thus for each j, 1 <= j <= n, the set X_j = A_j E + B_j = {A_j u + B_j | u in E} is a subset of E isometric to a uniformly scaled copy of itself. Question: --------- Find an algorithm in terms of the A_j and B_j that determines whether or not the sets X_1, ..., X_n have nonempty intersection. —Dan