  Given a finite abelian group $G$ and a natural number $t$, there are two
natural substructures of the Cartesian power $G^t$; namely, $S^t$ where $S$ is
a subset of $G$, and $x+H$ a coset of a subgroup $H$ of $G^t$. A natural
question is whether two such different structures have non-empty intersection.
This turns out to be an NP-complete problem. If we fix $G$ and $S$, then the
problem is in $P$ if $S$ is a coset in $G$ or if $S$ is empty, and NP-complete
otherwise; if we restrict to intersecting powers of $S$ with subgroups, the
problem is in $P$ if $\bigcap_{n\in\mathbb{Z} \mid nS \subset S} nS$ is a coset
or empty, and NP-complete otherwise. These theorems have applications in the
article [Spe21], where they are used as a stepping stone between a purely
combinatorial and a purely algebraic problem.
