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 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 phi: {[}X] -->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 cp satisfies the Lipschitz-type condition textbackslash{}textbackslash{}phi(x) - phi(x')textbackslash{}textbackslash{} R(s) and a constant C such that textbackslash{}textbackslash{}phi(x) - (L) over tilde(x) textbackslash{}textbackslash{} less than or equal to C for all x is an element of X. Suppose that X is a Delone set with an inflation symmetry, which is a real number eta > 1 such that eta X subset of or equal to X. rf X is a finitely generated Delone set, then eta must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates textbackslash{}eta'textbackslash{} less than or equal to eta; and if X is a Meyer set, then all algebraic conjugates textbackslash{}eta'textbackslash{} less than or equal to 1.