The Coven-Meyerowitz tiling conditions for 3 odd prime factors

• 成果类型: Article
• 署名作者: Laba, Izabella; Londner, Itay
• 署名单位: University of British Columbia; Weizmann Institute of Science
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01169-y
• 发表日期: 2023
• 页码: 365-470
• 关键词: spectral set conjecture; fugledes conjecture; vanishing sums; roots; holds; factorization; Integers; tiles

摘要

It is well known that if a finite set A subset of Z tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A circle plus B = Z(M) of a finite cyclic group. We are interested in characterizing all finite sets A subset of 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 vertical bar A vertical bar. We prove that the Coven-Meyerowitz tiling condition (T2) holds for all integer tilings of period M = (p(i) P-j P-k)(2), where p(i), p(j), p(k) are distinct odd primes. The proof also provides a classification of all such tilings.