## View abstract

### Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Tuesday, July 20, 16:00 ~ 16:30 UTC-3

## Tiling the integers with translates of one tile: the Coven-Meyerowitz tiling conditions for three prime factors

### Izabella Laba

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 $A\oplus B=\mathbb{Z}_M$ 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 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.