Geometric Models for Quasicrystals I. Delone Sets of Finite Type |
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Authors: | J C Lagarias |
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Institution: | (1) AT&T Labs, 180 Park Avenue, Florham Park, NJ 07932, USA jcl@research.att.com, US |
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Abstract: | This paper studies three classes of discrete sets X in
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 . 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
s
and
n
, respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map
and a constant C such that ||ϕ (x) - (x)|| for all x∈X .
Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that . 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.
Received May 9, 1997, and in revised form March 5, 1998. |
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