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 Izabella Laba (UBC), we proved that this is true for all sets tiling the integers with period $M = (pqr)^2$. In my talk I will discuss this problem and introduce some ideas from the proof.
Itay Londner