Pure point diffraction

Many important mathematical models of distributions of matter with pure point diffraction arise out of the cut and project formalism. As we show here, under some basic assumptions, the latter emerges very naturally from inherent data of pure point diffractive structures. Moreover, the internal space, whose physical meaning is often questioned, is the expression of a naturally occurring topology (the autocorrelation topology) that is part of the physical picture, though very different from the usual topology of space. We also indicate some of the recent progress in the geometrical properties that can be inferred from knowing that a uniformly discrete structure is pure point diffractive.