0 Lattice Substitution Systems and Model Sets

The paper studies ways in which the sets of a partition of a lattice in Rn become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in Rn gives rise to regular model sets (based on p-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the n−dimensional chair tiling and the sphinx tiling are pure point diffractive.