Geometric Models for Quasicrystals I. Delone Sets of Finite Type

Abstract

This paper studies three classes of discrete sets X in $\Bbb R$ n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism $\varphi : [X] \rightarrow {\Bbb Z}^s$ . A Delone set of finite 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 finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.'' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on $\Bbb R$ s and $\Bbb R$ n , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map $\tilde{L} : {\Bbb R}^n \rightarrow {\Bbb R}^s$ and a constant C such that ||ϕ (x) - $\tilde{L}$ (x)|| $\leq C$ for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that $\eta X \subseteq X$ . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | $\leq$ η; and if X is a Meyer set, then all algebraic conjugates | η ' | $\leq$ 1.