Dynamical systems of number-theoretic origin in the theory of aperiodic order

Regular model sets (a special class of cut and project sets), which go back to Yves Meyer (1972) in mathematics and to Peter Kramer (1982) in physics, form a versatile class of structures with amazing harmonic properties. These sets are also known as mathematical quasicrystals, and include the famous Penrose tiling with fivefold symmetry as well as its various generalisations to other non-crystallographic symmetries. They are widely used to model the structures discovered in 1982 by Dan Shechtman (2011 Nobel Laureate in Chemistry). More recently, also systems such as the square-free integers or the visible lattice points have been studied in this context, leading to the theory of weak model sets. This is an extension of the class of regular model sets that was also briefly considered by Meyer and by Schreiber in the 1970s, but has not seen any systematic investigation. Due to the connection with B-free integers and lattice systems, which are of renewed interest in the light of Sarnak's research program around M"obius orthogonality, weak model sets are now being studied in more detail by several groups. This talk will review some of the developments, and introduce important concepts from the field of aperiodic order, with focus on spectral aspects.