Bohr and Besicovitch almost periodic discrete sets and quasicrystals

Source abstract

A discrete set A A in the Euclidean space is almost periodic if the measure with the unit masses at points of the set is almost periodic in the weak sense. We investigate properties of such sets in the case when A − A A-A is discrete. In particular, if A A is a Bohr almost periodic set, we prove that A A is a union of a finite number of translates of a certain full–rank lattice. If A A is a Besicovitch almost periodic set, then there exists a full–rank lattice such that in most cases a nonempty intersection of its translate with A A is large.