[math-fun] Is the triangular array a disjoint union of proper similar ones?

Let w = primitive cube root of unity, and let E = {P(w) in C | P is an integer polynomial} be the triangular array in question, where C = complex plane, aka the Eisenstein integers. Definition: ----- A proper triangular array (PTA) is any proper subset A of E that is similar to E. ----- Any PTA A is of form A = a E + b for a and b any elements of E with |a| > 1. Question: ----- Does there exist a collection {A_j | j in J} of disjoint PTA's whose union is E ??? ----- ... where J is any index set —Dan