  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 spaces, 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 configuration 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 a follow-up paper that all tilings of period $(pqr)^2$, where
$p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven
and Meyerowitz.
