Cut-and-project sets and their ε-duals

Motivated by approximation and real analysis, Meyer introduced model sets (also called cut-and-project sets), which are used as mathematical models of quasicrystals. In his study, a central role was played by the -dual. The -dual of a lattice is the reciprocal lattice, and that of a cut-and-project set is contained by the diffraction pattern. Let [image omitted] be the cut-and-project set determined by locally compact Hausdorff Abelian groups [image omitted], lattice [image omitted] and a window [image omitted]. Then we prove [image omitted] by using the Baake-Lenz-Schlottmann measure dynamical system of point sets. Moreover, we characterize the symmetries of [image omitted] by the symmetries of W.