Discrete quasiperiodic sets with predefined local structure

Model sets play a fundamental role in the structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as a symmetry group a finite group GG, and there is a GG-cluster CC (union of orbits of GG) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of CC. We present an algorithm which leads from any GG-cluster CC directly to a multi-component model set QQ such that the arithmetic neighbours of any point x∈Qx∈Q are distributed on the sites of the translated copy x+Cx+C of CC. Our mathematical algorithm may be useful in quasicrystal physics.