A note on reduction of tiling problems

Abstract 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 N 1, …, N k ∈ ℕ the existence of an aperiodic tile in ℤ d−k × (ℤ/N 1ℤ × ⋯ × ℤ/N k ℤ) 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/N 1ℤ × ⋯ × ℤ/N k ℤ) for a suitable d, N 1,⋯, N k ∈ ℕ.