A note on reduction of tiling problems

We show that translational tiling problems in a quotient of ℤ^d can be effectively reduced or “simulated” by translational tiling problems in ℤ^d. In particular, for any d ∈ ℕ, k < d and N₁, …, N_k ∈ ℕ the existence of an aperiodic tile in ℤ^d−k × (ℤ/N₁ℤ × ⋯ × ℤ/N_kℤ) implies the existence of an aperiodic tile in ℤ^d. Greenfeld and Tao have recently disproved the well-known periodic tiling conjecture in ℤ^d for sufficiently large d ∈ ℕ by constructing an aperiodic tile in ℤ^d−k × (ℤ/N₁ℤ × ⋯ × ℤ/N_kℤ) for a suitable d, N₁,⋯, N_k ∈ ℕ.