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

—Dan

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