The Coven–Meyerowitz tiling conditions for 3 odd prime factors
Izabella Łaba, Itay Londner
Abstract
Abstract It is well known that if a finite set $$A\subset \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $$A\oplus B=\mathbb {Z}_M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:mrow> </mml:math> of a finite cyclic group. We are interested in characterizing all finite sets $$A\subset \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> that have this property. Coven and Meyerowitz (J Algebra 212:161–174, 1999) proposed conditions (T1), (T2) that are sufficient for A to tile, and necessary when the cardinality of A has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of | A |. We prove that the Coven–Meyerowitz tiling condition (T2) holds for all integer tilings of period $$M=(p_ip_jp_k)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , where $$p_i,p_j,p_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> are distinct odd primes. The proof also provides a classification of all such tilings.