The Coven–Meyerowitz tiling conditions for 3 odd prime factors
The Coven–Meyerowitz tiling conditions for 3 odd prime factors
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3 个奇素因子的 Coven-Meyerowitz 平铺条件
DOI：
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发表时间：
2021
影响因子：
3.1
通讯作者：
Itay Londner
中科院分区：
文献类型：
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作者：
I. Łaba;Itay Londner
众所周知，如果有限集 $$Asubset mathbb {Z}$$ A ⊂ Z 通过平移对整数进行平铺，则平移集必须是周期性的，因此平铺相当于有限循环群的分解 $$Aoplus B=mathbb {Z}_M$$ A ⊕ B = Z M 。我们感兴趣的是表征具有此属性的所有有限集 $$Asubset mathbb {Z}$$ A ⊂ Z。 Coven 和 Meyerowitz (J Algebra 212:161–174, 1999) 提出了 A 平铺的充分条件 (T1)、(T2)，并且当 A 的基数至多有两个不同的质因数时是必要的。他们还证明（T1）适用于所有有限的图块，无论大小。目前尚不清楚 (T2) 是否必须适用于所有平铺且对 | 的素因数数量没有限制。一个|。我们证明 Coven-Meyerowitz 平铺条件 (T2) 对于周期 $$M=(p_ip_jp_k)^2$$ M = ( p i p j p k ) 2 的所有整数平铺成立，其中 $$p_i,p_j,p_k$$ p i , p j , p k 是不同的奇素数。该证明还提供了所有此类平铺的分类。
It is well known that if a finite set $$Asubset mathbb {Z}$$ A ⊂ Z tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $$Aoplus B=mathbb {Z}_M$$ A ⊕ B = Z M of a finite cyclic group. We are interested in characterizing all finite sets $$Asubset mathbb {Z}$$ A ⊂ Z 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 $$M=(p_ip_jp_k)^2$$ M = ( p i p j p k ) 2 , where $$p_i,p_j,p_k$$ p i , p j , p k are distinct odd primes. The proof also provides a classification of all such tilings.