Caltech/UCLA Joint Analysis Seminar
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=ZM 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 behaviour 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. (We are currently finalizing the even case.) In my talk I will discuss this problem and introduce the main ingredients in the proof.