2022/07/24 by Izabella Łaba, Laba, Izabella, Itay Londner +1 · 1 citation
Computer Science · Engineering · Materials Science · #05B45 #11B75 #20K01 #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Quasicrystal Structures and Properties #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2207.11809
openalex publication_date 2022/07/24 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We consider finite sets A⊂ℤ tiles the integers by translations. By periodicity, any such tiling is equivalent to a factorization A⊕ B=ℤM of a finite cyclic group. Building on por previous work, we prove that a tentative characterization of finite tiles proposed by Coven and Meyerowitz holds for all integer tilings of period M=(pipjpk)2, where pi,pj,pk are distinct primes. This extends the main result of [15] (Invent. Math. 2023), where we assumed that M is odd. We also improve parts of the argument from [15]. We have split the earlier (70-page) version into two papers. The current version (49 pages) is the first of the two. The main result is the same as in the previous version: we prove (T2) in the 3-prime even case. The second paper will be posted shortly as a new submission. It will have a new main result where we prove (T2) for a new class of tilings (proved very recently, not included in v1 of this paper). Splitting-related results from the earlier 70-page version of this paper have been moved there.