You are right, Jim. Actually, it's a kind of fun puzzle to find a partition E = U E_j of E as the disjoint union of cosets E_j of distinct |a_j|, i.e., the magnification factors |a_j| satisfy : j ≠ k => |a_j| ≠ |a_k|. Of course, the densities d_j = 1 / |a_j|^2 will be numbers of the form 1/K*K + K*L + L*L) for some K,L in Z, such that (*) Sum d_j = 1. Conjecture: Maybe a converse is true (?): ----- Whenever a set {d_j} of rationals of the form 1/(K*K + K*L + L*L) sums to 1, then there exists a partition of E into E_j as in (*). ----- —Dan Jim Propp wrote: ----- Dan, did you mean to specify that the dilation factors |a_j| should all be distinct? Because if you don’t require this, it seems to me that there are numerous examples; e.g., for a fixed a, take all the distinct cosets of a E. Jim Propp On Wed, May 15, 2019 at 6:58 PM Dan Asimov <dasimov@earthlink.net> wrote:

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