  The theory of regular model sets is highly developed, but does not cover
examples such as the visible lattice points, the k-th power-free integers, or
related systems. They belong to the class of weak model sets, where the window
may have a boundary of positive measure, or even consists of boundary only. The
latter phenomena are related to the topological entropy of the corresponding
dynamical system and to various other unusual properties. Under a rather
natural extremality assumption on the density of the weak model set we
establish its pure point diffraction nature. We derive an explicit formula that
can be seen as the generalisation of the case of regular model sets.
Furthermore, the corresponding natural patch frequency measure is shown to be
ergodic. Since weak model sets of extremal density are generic for this
measure, one obtains that the dynamical spectrum of the hull is pure point as
well.

Dynamical Systems;Functional Analysis