Self-similar Delone sets and quasicrystals

In this paper we answer the question, whether any Delone set Lambda subset of R-n, invariant under quasiaddition of Berman and Moody, can be identified with a cut and project quasicrystal. For any such set Lambda, we find an acceptance window Omega, which is bounded but has only convex interior. The cut and project quasicrystal Sigma(Omega) is then identified with an affine image of Lambda. Constructive methods used in the paper, allow one, in principle, to put bounds on Omega from a given fragment of a Delone set.