Tiling lattices in Z^2

A tiling of Z^n is a collection of sublattices L_1,...,L_m of Z^n together with integer-valued vectors v_1,...,v_m such that the translates v_1+L_1,...,v_m+L_m are disjoint and cover ("tile") the whole Z^n. The question we address is, given a tiling, do there exist two different indices k,l, such that L_k=L_l? It is true for n=1, there are easy counterexamples if n>2. The case n=2 is open. We show that a tiling of Z^2 such that L_k!=L_l for k!=l must contain a lattice of index at least 36. The tools are elementary: generating function and multivariable residue calculus. This is a joint work with D. Nguyen and S. Robins.