  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.
