Splitting for integer tilings

Abstract

We consider translational integer tilings by finite sets $A\subset\mathbb{Z}$. We introduce a new method based on \emph{splitting}, together with a new combinatorial interpretation of some of the main tools from our earlier work. We also use splitting to prove the Coven-Meyerowitz conjecture for a new class of tilings $A\oplus B=\mathbb{Z}_M$. This includes tilings of period $M=p_1^{n_1}p_2^{n_2}p_3^{n_3}$ with $p_1>p_2^{n_2-1}p_3^{n_3-1}$, and tilings of period $M=p_1^{n_1}p_2^2p_3^2p_4^2$ with $p_1>p_2p_3p_4$, where $p_1,p_2,p_3,p_4$ are distinct primes and $n_1,n_2,n_3\in\mathbb{N}$. This is the second one of the two papers replacing version 1 of arXiv:2207.11809 (the first one is available as arXiv:2207.11809 v2). The main results of this paper (Theorem 1.2, Corollaries 1.4 and 1.5) and the intermediate results in Section 4.2 are all new and did not appear previously in arXiv:2207.11809 v1 or anywhere else. The material in Sections 3, 4.1, and 5 (splitting and the splitting formulation of the slab reduction) did appear in arXiv:2207.11809 v1 and has been removed from arXiv:2207.11809 v2. The results in Section 7 were included in arXiv:2207.11809 v1 and have been removed from arXiv:2207.11809 v2; the proofs are shorter and (we hope) more readable.