Dynamical Spectrum and Diffraction

The diffraction spectrum of a point pattern is very closely related to the dynamical spectrum of an associated dynamical system. This dynamical spectrum is invariant under topological conjugacy and measurable conjugacy, and in particular under a large class of shape deformations. Using measure theory and topology, we construct a pure-point diffractive set, with finite local complexity, that is not a Meyer set. This provides a counterexample to a famous conjecture of Lagarias.