It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered “standard” tiling complement. It is not known whether this behavior persists for all tilings with no restrictions on the number of prime factors of M.

In joint work with Itay Londner, we proved that this is true when M=(pqr)^2 is odd. (The even case is almost finished as well.) In my talk, I will discuss this problem and introduce the main ingredients of the proof.