Quasicrystals and almost periodicity

We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let [InlineMediaObject not available: see fulltext.] denote the space of uniformly discrete subsets of the Euclidean space. Let [InlineMediaObject not available: see fulltext.] denote the elements of [InlineMediaObject not available: see fulltext.] that admit an autocorrelation measure. A Patterson set is an element of [InlineMediaObject not available: see fulltext.] such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S [InlineMediaObject not available: see fulltext.] is a Patterson set if and only if S is almost periodic in ([InlineMediaObject not available: see fulltext.],[InlineMediaObject not available: see fulltext.]), where [InlineMediaObject not available: see fulltext.] denotes the Besicovitch topology. Let χ be an ergodic random element of [InlineMediaObject not available: see fulltext.]. We prove that χ is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets. © Springer-Verlag 2005.