UBC Math Department Colloquium: Izabella Laba

A set $A\subset\mathbb{Z}$ tiles the integers by translations if there is a set $T\subset\mathbb{Z}$ such that every integer
$n\in\mathbb{Z}$ has a unique representation $n=a+t$ with $a\in A$ and $t\in T$. It is well known that the translation set in any such tiling must be periodic, so that the tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group.

The main open question regarding integer tilings is the Coven-Meyerowitz conjecture, providing a tentative characterization of
finite tiles. The conjecture, if true, would imply that each of the sets $A$ and $B$ in any tiling $A\oplus B=\mathbb{Z}_M$ can be replaced by a highly ordered "standard" tiling complement. Coven and Meyerowitz (1998) proved that these conditions hold for tiles $A$ whose cardinality has at most two distinct prime factors. The general case remains open.

In the last few years, Itay Londner and I proved the conjecture for several new classes of tilings, including all tilings of period $M=(pqr)^2$, where $p, q, r$ are distinct primes. I will discuss the main ideas of the proof, as well as implications for other tiling questions such as estimates on the minimal tiling period.