  We show that translational tiling problems in a quotient of $\mathbb{Z}^d$
can be effectively reduced or ``simulated'' by translational tiling problems in
$\mathbb{Z}^d$. In particular, for any $d \in \mathbb{N}$, $k < d$ and
$N_1,\ldots,N_k \in \mathbb{N}$ the existence of an aperiodic tile in
$\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times
\mathbb{Z} / N_k \mathbb{Z})$ implies the existence of an aperiodic tile in
$\mathbb{Z}^d$. Greenfeld and Tao have recently disproved the well-known
periodic tiling conjecture in $\mathbb{Z}^d$ for sufficiently large $d \in
\mathbb{N}$ by constructing an aperiodic tile in $\mathbb{Z}^{d-k} \times
(\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$
for suitable $d,N_1,\ldots,N_k \in \mathbb{N}$.
