Cut-and-project sets and their -duals

Abstract

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 ϵ-dual. The ϵ-dual of a lattice is the reciprocal lattice, and that of a cut-and-project set is contained by the diffraction pattern. Let be the cut-and-project set determined by locally compact Hausdorff Abelian groups , lattice and a window . Then we prove by using the Baake–Lenz–Schlottmann measure dynamical system of point sets. Moreover, we characterize the symmetries of by the symmetries of W.

Acknowledgements

The second author is supported by COE ‘Exploring New Science by Bridging Particle–Matter Hierarchy,’ Graduate School of Science, Tohoku University.