A finite set of integers A tiles the integers by translations if
$\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have
$A\oplus B=\mathbb {Z}_M$ for some
$M\in \mathbb {N}$ and
$B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials
$A(X)$ and
$B(X)$ associated with A and B.
In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group
$\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period
$(pqr)^2$, where
$p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].