Caltech/UCLA Joint Analysis Seminar

Series: Caltech/UCLA/USC Joint Analysis Seminar Series

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

Izabella Laba, Department of Mathematics, University of British Columbia,

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.

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