Geometric models for quasicrystals I. Delone sets of finite type

Abstract

This paper studies three classes of discrete sets X in ℝn 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 φ: [X] → ℤs. 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 ℝs and ℝn, respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map L̃: ℝn → ℝs and a constant C such that ∥φ(x) - L̃(x)∥ ≤ 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 ηX ⊆ 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′| ≤ η; and if X is a Meyer set, then all algebraic conjugates \η′\ ≤ 1.