## 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

#### University of British Columbia, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak7408ffd993ab6f17be03ed24689dfe98').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy7408ffd993ab6f17be03ed24689dfe98 = '&#105;l&#97;b&#97;' + '&#64;'; addy7408ffd993ab6f17be03ed24689dfe98 = addy7408ffd993ab6f17be03ed24689dfe98 + 'm&#97;th' + '&#46;' + '&#117;bc' + '&#46;' + 'c&#97;'; var addy_text7408ffd993ab6f17be03ed24689dfe98 = '&#105;l&#97;b&#97;' + '&#64;' + 'm&#97;th' + '&#46;' + '&#117;bc' + '&#46;' + 'c&#97;';document.getElementById('cloak7408ffd993ab6f17be03ed24689dfe98').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy7408ffd993ab6f17be03ed24689dfe98 + '\'>'+addy_text7408ffd993ab6f17be03ed24689dfe98+'<\/a>';

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.

Joint work with Itay Londner (University of British Columbia, Canada).