id: 01271043
dt: j
an: 01271043
au: Lagarias, J.C.
ti: Geometric models for quasicrystals. I. Delone sets of finite type.
so: Discrete Comput. Geom. 21, No.2, 161-191 (1999).
py: 1999
pu: Springer-Verlag, New York, NY
la: EN
cc: 
ut: finitely generated Delone set
ci: 
li: doi:10.1007/PL00009413
ab: Summary: This paper studies three classes of discrete sets $X$ in
    ${\bbfR}^n$ 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 $φ: [X]\to {\bbfZ}^s$. 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 $\|φ({\bold x})- φ({\bold x}’)\|<
    C\|{\bold x}-{\bold x}’\|$ for ${\bold x},{\bold x}’\in X$, where
    the norms $\|\cdot\|$ are Euclidean norms on ${\bbfR}^s$ and
    ${\bbfR}^n$, respectively. Meyer sets are characterized as the subclass
    of Delone sets of finite type for which there is a linear map
    $\widetilde L: {\bbfR}^n\to {\bbfR}^s$ and a constant $C$ such that
    $\|φ({\bold x})- \widetilde L({\bold x})\|\le C$ for all ${\bold x}\in
    X$. Suppose that $X$ is a Delone set with an inflation symmetry, which
    is a real number $η> 1$ such that $ηX\subseteq X$. 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 $|η’|\le η$; and if $X$ is a Meyer set, then all
    algebraic conjugates $|η|\le 1$.
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