Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in R n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X ? X. A nitely generated Delone set is one such that the abelian group X ? X] generated by X ? X is nitely generated, so that X ? X] is a lattice or a quasilattice. For such sets the abelian group X] is nitely generated, and by choosing a basis of X] one obtains a homomorphism : X]!Z s. A Delone set of nite type is a Delone set X such that X ? X is a discrete closed set. A Meyer set is a Delone set X such that X ? X is a Delone set. Delone sets of nite type form a natural class for modelling quasicrystalline structures, because the property of being a Delone set of nite type is determined by \local rules." That is, a Delone set X is of nite type if and only if it has a 20 nite number of neighborhoods of radius 2R, up to translation, where R is the relative denseness constant of X. Delone sets of nite type are also characterized as those nitely generated Delone sets such that the map satisses the Lipschitz-type condition jj(x) ? (x 0)jj < Cjjx ? x 0 jj for x; x 0 2 X, where the norms jj jj are Euclidean norms on R s and R n , respectively. Meyer sets are characterized as the subclass of Delone sets of nite type for which there is a linear map ~ L : R n !R s and a constant C such that jj(x) ? ~ L(x)jj C for all x 2 X. Suppose that X is a Delone set with an innation symmetry, which is a real number > 1 such that X X. If X is a nitely generated Delone set, then must be an algebraic integer; if X is a Delone set of nite type then in addition all algebraic conjugates j 0 j ; and if X is a Meyer set then all algebraic conjugates j 0 j 1.