A note on reduction of tiling problems

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 $N_1,\dots ,N_k\in \mathbb{N}$ the existence of an aperiodic tile in ${\mathbb{Z}}^{d-k}\times (\mathbb{Z}\mathrm{/}{N}_1\mathbb{Z}\times \dots \times \mathbb{Z}\mathrm{/}{N}_k\mathbb{Z})$ 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 ${\mathbb{Z}}^{d-k}\times (\mathbb{Z}\mathrm{/}{N}_1\mathbb{Z}\times \dots \times \mathbb{Z}\mathrm{/}{N}_k\mathbb{Z})$ for suitable $d,N_1,\dots ,N_k\in \mathbb{N}$.