Event № 135
We will start by recalling the definition of diffraction of a quasi-crystal explained in last week’s talk. The majority of the talk is then devoted to the question how to construct explicit examples of quasi-crystals with pure point diffraction.
We will introduce cut-and-project schemes and define the notion of a model set. It turns out that model sets are examples of quasi-crystals with “mostly” pure-point diffraction, and in some sense they are the only such examples (Meyer's theorem).
Using the dynamical system on the hull explained in last week’s talk, we will derive an explicit formula for the diffraction of a regular model set. The key ingredient is Schlottmann’s torus parametrisation, which provides a measurable isomorphism between the dynamical system on the hull of a regular model set and an almost homogeneous system, which in the case of R^n is simply an irrational rotation on a torus.
Time permitting we will discuss possible generalizations to non-commutative (and in particular arithmetic) quasi-crystals in the sense of our recent work with Björklund and Pogorzelski.