Discrete Comput Geom 21:161–191 (1999) Discrete & Computational

Geometry© 1999 Springer-Verlag New York Inc.

Geometric Models for QuasicrystalsI. Delone Sets of Finite Type

J. C. Lagarias

AT&T Labs, 180 Park Avenue,Florham Park, NJ 07932, USAjcl@research.att.com

Abstract. This paper studies three classes of discrete setsX in Rn which have a weaktranslational order imposed by increasingly strong restrictions on their sets of interpointvectorsX − X. A finitely generated Delone setis one such that the abelian group [X − X]generated byX − X is finitely generated, so that [X − X] is a lattice or a quasilattice. Forsuch sets the abelian group [X] is finitely generated, and by choosing a basis of [X] oneobtains a homomorphismϕ: [X] → Zs. A Delone set of finite typeis a Delone setX suchthat X − X is a discrete closed set. AMeyer setis a Delone setX such thatX − X is aDelone 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.” Thatis, a Delone setX is of finite type if and only if it has a finite number of neighborhoods ofradius 2R, up to translation, whereR is the relative denseness constant ofX. Delone sets offinite type are also characterized as those finitely generated Delone sets such that the mapϕ

satisfies the Lipschitz-type condition‖ϕ(x)−ϕ(x′)‖ < C‖x− x′‖ for x, x′ ∈ X, where thenorms‖ · ‖ are Euclidean norms onRs andRn, respectively. Meyer sets are characterizedas the subclass of Delone sets of finite type for which there is a linear mapL: Rn → Rs

and a constantC such that‖ϕ(x)− L(x)‖ ≤ C for all x ∈ X.Suppose thatX is a Delone set with an inflation symmetry, which is a real numberη > 1

such thatηX ⊆ X. If X is a finitely generated Delone set, thenη must be an algebraicinteger; ifX 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.

1. Introduction

The discovery of quasicrystalline materials in 1984 has generated much theoretical andexperimental work [10], [29], [30], [56], [77]. These are materials which have X-ray

162 J. C. Lagarias

diffraction spectra that have sharp spots (“Bragg peaks”) indicating long range order oftheir atomic structure under translations, but which exhibit forbidden symmetries indi-cating that they cannot have crystallographic atomic structure. The list of quasicrystallinematerials now includes a few types of thermodynamically stable “perfect quasicrystals”which have structure as uniform as crystals, as well as a much larger variety of quasicrys-talline structures with less perfect order [20]. There are a number of different structuralmodels proposed for quasicrystalline materials, ranging from random tiling models [22],[23] to models based on cut-and-project sets [37], [43], [44]. Indeed such structures canexhibit diffraction patterns of approximately the right kind [12], [24], [29], [53]. Thisseries of papers is motivated by the problem of finding atomic models for such materials,that can describe both thermodynamically stable quasicrystals and more disordered qua-sicrystalline structures. In particular, we consider sets with weaker kinds of long-rangeorder under translation that would not necessarily be manifested by “Bragg peaks” intheir diffraction spectrum.

The concept of a Delone set is a mathematical abstraction for the positions of atomsin a solid state material.

Definition 1.1. A discrete setX is a Delone set(or Delaunay setor (r, R)-set) if itsatisfies the following two conditions:

(i) Uniform Discreteness. There is a valuer such that each open ball of radiusrcontains at most one point ofX.

(ii) Relative Denseness. There is a valueR such that each closed ball of radiusRcontains at least one point ofX.

The maximal value ofr is the packing radius forX using equal spheres centered atits points, and the minimal value ofR is the covering radius ofX using equal spherescentered at its points to coverRn.

The notion of Delone sets as the fundamental object of study in crystallography wasproposed by the Russian school in the 1930s, see [8], [9], [19], and p. 25 of [71]. Deloneset models for quasicrystals have been studied by a number of authors, e.g., Danzer [5,p. 571] and Klitzing and Baake [35].

In this paper we consider three classes of discrete geometric structures inRn, whichconsist of Delone setsX with some translational order imposed by restrictions on theirset of interpoint vectorsX−X. It is natural to consider restrictions on the set of interpointvectorsX − X, first because the diffraction pattern is determined by the setX − X, andsecond because restrictions on local arrangements of atoms appear to be imposed insome materials by local energy minimization.1

The three classes of Delone sets that we study are specified by increasingly strongrestrictions on their sets of interpoint vectorsX − X, as follows.

1 The resultant of the forces acting on each atom must be zero (local stability). The electromagnetic forceis long range, but the dominant effects on one atom come from nearby atoms due to approximate cancellationof forces from atoms further away.

Geometric Models for Quasicrystals, I 163

Definition 1.2. Let X be a Delone set.

(i) X is afinitely generated Delone setif the abelian group

[X − X] = Z[x− x′ : all x, x′ ∈ X] (1.1)

is finitely generated.(ii) X is aDelone set of finite typeif X − X is a discrete closed subset ofRn, i.e.,

the intersection ofX − X with any closed ball is a finite set.(iii) X is aMeyer setif X − X is a Delone subset ofRn.

We show that class (i) includes class (ii), and class (ii) obviously includes class (iii).The terminology “Delone set of finite type” is motivated by the fact that Delone setsof finite type have only finitely many types of local neighborhoods of radius 2R undertranslations, see Section 2. More precisely (ii) defines Delonesets of finite type undertranslations, to distinguish them from the larger class ofDelone sets of finite type underisometriesconsidered in [11] and [39]. The class of Meyer sets was introduced by Meyer[53] under the name “quasicrystal”; they are now generally called Meyer sets, see [38],[54], and [55]. Definition (iii) of a Meyer set is not the usual one, but in [38] it is shownequivalent to Meyer’s original definition thatX − X ⊆ X + F , for some finite setF .Meyer [53] observes that all cut-and-project sets are Meyer sets, and presents evidencethat they form a suitable class to be termed quasicrystals. We argue here that the moreinclusive class of Delone sets of finite type is the natural general class of sets to considerin making models for the atomic structure of quasicrystalline materials. However, bothof these classes include sets that are not quasicrystalline in the sense of possessing awell-defined diffraction spectrum.Various examples that illustrate these points appear inSection 5.

A Delone setX is finitely generated if and only if the abelian group

[X] := Z[x : x ∈ X] (1.2)

is finitely generated. We view this property as a weak form of long-range order undertranslations2 of the elements ofX. A finitely generated abelian group [X] that spansRn iscalled aquasilattice; it is called alattice if it is discretely embedded inRn. Quasilatticeshave been introduced as a unifying concept in studying the symmetries of quasicrystals,see [42], [50], and [60]. Therank of X is the number of generators of [X] as a freeabelian group. (All finitely generated subgroups ofRn are free.)

Definition 1.3. Let X be a finitely generated Delone set withrank(X) = s, and choosea basis of [X], say

[X] = Z[v1, v2, . . . , vs]. (1.3)

Theaddress mapϕ: [X] → Zs associated to this basis is

ϕ

(s∑

i=1

ni vi

)= (n1,n2, . . . ,ns). (1.4)

2 See the note on terminology at the end of the introduction.

164 J. C. Lagarias

The address map is specified by the choice of basis of [X], so it is determined upto left-multiplication by an element ofGL(s,Z). The structure of a finitely generatedDelone set is to some extent analyzable by studying its image inRs under the addressmap. Using the address map the points ofX can be coordinatized as a subsetY ofthe integer latticeZs regarded as embedded inRs, with s = rank(X), and there is alinear projectionψ : Rs → Rn with ψ(Y) = X. The address map describesX usings“ internal dimensions.” In this coordinatization alls dimensions are on an equal footing;there seems no natural way to selectn of these dimensions as “physical dimensions.”

The class of finitely generated Delone sets is the largest class of sets on which anaddress map is well defined. It seems to be too large a class to be interesting, butmembership in it does put algebraic restrictions on inflation symmetries, as described inSection 4. Various interesting subclasses of such sets are obtained by putting restrictionson the address map which require that points that are close inX have addresses which arenot too far apart in the address spaceRs. The most important subclass of such sets seemsto be Delone sets of finite type, which we show are characterized by a Lipschitz-typecondition on the address map.

The class of Delone sets of finite type appears to be a natural unifying concept inclassifying quasicrystalline structures. In Section 2 we show that Delone sets of finitetype form the largest class of Delone sets that can be described by “local rules undertranslations.” That is, we show that a Delone setX is of finite type if and only ifXhas only a finite number of different local neighborhoods of radius 2R around its pointsx ∈ X, up to translation (Theorem 2.1). This result shows that the property of being aDelone set of finite type is enforceable by local rules under translations, in the sense ofpart II, see also [42] and [45]. It also follows that a Delone setX of finite type inRn isalways finitely generated, andrank(X) is bounded by the number of points inX− X oflength at most 2R.

One way to model the atomic structure of quasicrystalline materials which have X-raydiffraction patterns which exhibit crystallographically forbidden symmetries is throughthe use of sets described using a finite number of extra “internal dimensions,” see [50].Certain cut-and-project sets provide examples; they are obtained as projections of part ofa higher-dimensional lattice, see Section 3. To get a Delone set inR3 having a diffractionpattern having “Bragg peaks” located at points with icosahedral symmetry, one canproject from a suitable six-dimensional lattice. The “local rules under translations” resultabove for Delone sets of finite type shows that finiteness of the number of “internaldimensions” describing a set follows as a consequence of finiteness of the number ofallowable local configurations under translations. In part II we strengthen this result byshowing that finiteness of the number of “internal dimensions” can also be forced bylocal rules under isometries, for a large class of structures.

Tiling models of quasicrystals naturally produce Delone sets of finite type. Consider atiling ofRn which consists of translates of finitely many types of prototiles. We associate aDelone setX to the tiling by marking a finite set of points in each prototile (“decorations”),see, for example, [35]. If there are only a finite number of different ways two prototilesmay be adjacent in this tiling, thenX is necessarily a Delone set of finite type, andconversely (Corollary 2.1). In particular, random tiling models of quasicrystals in thesense of Henley [23, Section 2.3] yield Delone sets of finite type.

In Section 2 we give several characterizations of Delone sets of finite type. We show

Geometric Models for Quasicrystals, I 165

that they are characterized by either of the following conditions:

(1) The address mapϕ satisfies the Lipschitz-type condition

‖ϕ(x)− ϕ(x′)‖ ≤ C‖x− x′‖ for x, x′ ∈ X,

where the norms are Euclidean norms onRs andRn, respectively.(2) The tiling ofRn by the Voronoi domains induced byX (with the points ofX

marked) contains finitely many translation-equivalence classes of marked Voronoidomains.

(3) The Delaunay tiling ofRn induced byX contains finitely many translation-equivalence classes of Delaunay domains.

In Section 2 we also show that Delone sets of finite type inRn have bounds ontheir number of distinct local configurations under translations. LetN∗X(T) be theatlas counting function which counts the number of translation-inequivalent patches(X − x) ∩ B(0; T) of radiusT over allx ∈ X. Theconfigurational entropy Hc(X) of asetX in Rn is defined by

Hc(X) := lim supT→∞

log N∗X(T)Tn

, (1.5)

in analogy with definitions in [41]. We show that any Delone set of finite typeX hasfinite configurational entropy, i.e., there exists a constantC depending onX such that

N∗X(T) ≤ exp(CTn). (1.6)

This is best possible; in Section 3 we give an example of a Meyer setX in Rn withpositive configurational entropy

N∗X(T) ≥ exp(C′Tn) as T →∞,

whereC′ is a positive constant.In Section 3 we study Meyer sets viewed as a subclass of Delone sets of finite type.

These sets have been extensively studied and characterized by Meyer [53] and Moody[54], [55]. We show that they are exactly those Delone setsX of finite type whose addressmap is an approximate linear mapping onX. That is, they are those setsX such thatthere exists a one-to-one linear mapL: Rn→ Rs such that

‖ϕ(x)− L(x)‖ ≤ C, all x ∈ X. (1.7)

Meyer previously showed that a setX is a Meyer set if and only ifeveryhomomorphismψ : [X] → Rd for 1≤ d <∞ is an approximate linear mapping onX, see Theorem 9.1of [29]. Our small addition here is to observe that it suffices to check this property forthe single mapϕ. The criterion (1.7) is useful in determining whether a setX is a Meyerset, and we apply it in Section 5. We also show that every Meyer setX is contained insome cut-and-project setY of dimension at mostrank(X).

We introduce a relation oflocal derivability between Delone sets which parallelsa notion for tilings introduced by Baake et al. [3]. LetY be a Delone set which is

166 J. C. Lagarias

locally derivable fromX. We show that ifX is a Delone set of finite type, then so isY(Corollary 2.1c), and ifX is a Meyer set, then so isY (Corollary 3.1b).

In Section 4 we study Delone setsX that have aninflation symmetryη, which is areal numberη > 1 such thatηX ⊆ X. This is not a true symmetry, sinceX\ηX containsinfinitely many points. Delone sets having an inflation symmetry can be obtained fromself-similar tiling models of quasicrystals as sets of “control points,” see [18], [32], [53],and [74]. We show that:

(1) If X is a finitely generated Delone set, thenη is an algebraic integer.(2) If X is a Delone set of finite type, thenη is an algebraic integer all of whose

algebraic conjugatesη′ satisfy|η′| ≤ η.(3) If X is a Meyer set, thenη is an algebraic integer all of whose algebraic conjugates

η′ satisfy|η′| ≤ 1.

The result (3) was originally proved by Meyer [53, Theorem 6]. These results are ap-parently best possible in the sense that ifη has the stated property, then there exists anappropriate Delone setX having the symmetryηX ⊆ X. This has been demonstratedfor cases (1) and (3), and has been checked for some extremal examples in case (2).

In Section 5 we study one-dimensional Delone sets of finite type. Such sets correspondexactly to tilings of the lineR using a finite set of prototiles which are intervals ofdifferent lengths. We study special examples that are constructed using one-dimensionalsymbolic dynamical systems. These examples include some Meyer sets which might notbe considered “quasicrystals.” We also discuss diffraction spectra of one-dimensionalsets. It is not known whether there exist (minimal) non-Meyer sets which contain somediscrete component in their diffraction spectrum.

The deepest problem about quasicrystalline materials concerns finding physical mech-anisms that lead to noncrystallographic ordered structures.3 This paper does not addressthis question. However, we note that many recent studies of quasicrystalline structurepropose “clusters” or “local rules” as representing locally energy-minimizing finite ar-rangements of atoms at low temperature, and then propose decorations or “matchingrules” for how clusters may overlap or abut each other as producing quasicrystallinestructures, see [17], Section 2.2 of [23], Section 6 of [28], and [45]. The resulting struc-tures are Delone sets of finite type.

The viewpoint of this paper is that Delone sets of finite type form a suitable classof sets that encompass all structures which might reasonably be called quasicrystalline.The narrower class of Meyer sets has previously been proposed as such a class [29]–[32]. However self-affine tilings ofRn which have an inflation matrix whose largesteigenvalue is not a Pisot or Salem number correspond (by picking “control points” intiles) to Delone sets of finite type which are not Meyer sets. There exist such sets thathave long-range order under translations in the sense of having a well-defined diffractionmeasure. That is, they have a unique autocorrelation measure in the sense of Hof [24],[26]; the diffraction measure is the Fourier transform of the autocorrelation measure.These diffraction measures are not known to contain any discrete component except at0.It may be that there exist Delone sets of finite type which are not Meyer sets which have

3 Even the problem of giving a physical explanation for crystallographic order is not resolved, see [63]and [65].

Geometric Models for Quasicrystals, I 167

the quasicrystalline property of having some “Bragg peaks” in their diffraction patterns;this is currently an open question, see Section 5.

Various special cases of the mathematical framework developed here are implicit inprevious work. The notion of finite type for tilings appears under the name “finite numberof local patterns” in [74, p. 699] where earlier references are cited. Robinson [67] definesa notion of “tiling of finite type” which corresponds to a tiling which satisfies some finiteset of “local rules” under translations. Analogues of the address map have been previouslyintroduced in special cases for tiling models for quasicrystals, in studying local rules forsuch tilings. For example, it appears as the “lifting function” in [45], see also [31] and[42]. For cut-and-project sets it appears as “perp-space coordinates” in Section 3.1.1 of[23]. Danzer [5] has long advocated the study of sets describable by local conditions.Danzer and Dolbilin [7] recently introduced a notion of Delone graph which is analogousto “local rules under isometries.” They prove an analogue of Lemma 2.2, which is furthersharpened in [6].

Note on Terminology. The concept of “quasicrystal” does not have a generally agreedon definition in the literature. In the physics literature [77] it typically refers to materialswhose X-ray diffraction patterns exhibit some “Bragg peaks” with noncrystallographicsymmetries; thus crystals are not considered quasicrystals. In this paper we do not definethe term “quasicrystal” but the various classes of sets we define are inclusive: we viewideal crystals as a special kind of “quasicrystal.” In the physics literature “long-rangeorder” refers to the presence of Bragg peaks in a diffraction spectrum. Analogously tothis we consider a Delone setX to havestrong long-range order under translationsif[X] is a quasilattice andX has a well-defined diffraction measure in the sense of Hof[24], [26] which contains some discrete spectrum whose points spanRn. We considera Delone setX to havelong-range order under translationsif [ X] is a quasilattice andX has a well-defined diffraction measure, i.e.,X has a unique autocorrelation measure.Finally, we consider a Delone setX to haveweak long-range order under translationsif [ X] is a quasilattice, i.e., ifX is a finitely generated Delone set; this is a mathematicalnotion of long-range order that may not be easily detectable physically.

Notation. ‖ · ‖ denotes the Euclidean norm onRn. B(x; T) is the closed ball of radiusT aroundx ∈ Rn, i.e., B(x; T) := {y : ‖y − x‖ ≤ T}. The symbolC designates apositive constant, which may be different at different occurrences.

2. Delone Sets of Finite Type

A Delone setX is of finite typeif X − X is a discrete closed set, i.e., each ballB(0; T)contains only finitely many points ofX− X. In this section we characterize Delone setsof finite type in several ways and establish some of their properties.

Lemma 2.1. If X is a Delone set of finite type, then any Delone set Y with Y⊆ X is aDelone set of finite type.

168 J. C. Lagarias

Proof. The setY − Y ⊆ X − X, hence each ballB(0; T) contains only finitely manypoints ofY − Y.

It is useful to have a quantitative measure of the discreteness of the closed setX− X.

Definition 2.1. For a Delone setX, its distance counting function DX(T) is

DX(T) := #{y ∈ X − X : ‖y‖ ≤ T}. (2.1)

A Delone setX is of finite type if and only ifDX(T) is finite for all T > 0.

We now prove a “local rules” criterion for a Delone set to be of finite type, which alsoestablishes that such sets are finitely generated.

Theorem 2.1. If X is a Delone set with relative denseness constant R, then X is aDelone set of finite type if and only if

DX(2R) = #{(X − X) ∩ B(0;2R)} is finite. (2.2)

Any Delone set X of finite type is finitely generated, with

rank(X) ≤ DX(2R). (2.3)

To prove Theorem 2.1 we use the following “stepping-stone” property of Delone sets.

Lemma 2.2. Let X be a Delone set inRn with parameters(r, R). Given any two pointsx, x′ ∈ X there exists a chain of points

x = x0, x1, x2, . . . , xm = x′ (2.4)

in X such that

(i) ‖xi − xi−1‖ ≤ 2R, for 1≤ i ≤ m.(ii) m≤ (4R/r 2)‖x− x′‖.

Proof. We construct the sequence recursively as follows. Givenxi , if ‖xi − x′‖ ≤ 2Rsetm= i + 1 and stop. Otherwise consider the closed ball

Bi := B

(xi + R

x′ − xi

‖x′ − xi ‖; R)

of radiusR which has the property thatxi is on its boundary, andxi is the furthest pointin this ball fromx′. This ball Bi necessarily contains another point ofX. To show thiswe consider forε > 0 the translated ball

Bi (ε) := B

(xi + (R+ ε) x′ − xi

‖x′ − xi ‖; R).

It contains some point ofX by the Delone property, call itx(ε), andx(ε) 6= x sincexi 6∈ Bi (ε) for ε > 0. SinceX is discrete, some point occurs infinitely often among the

Geometric Models for Quasicrystals, I 169

points{x(1/n) : n = 1,2, . . .}, call it xi+1, and it is inBi because its distance fromBi isless than 1/n for infinitely manyn. Thus (i) holds. Next, the Delone property says that

‖xi+1− xi ‖ ≥ r.

However,xi is the furthest point in the ballBi from x′, so this condition slices a sphericalcap off Bi , and there is a constantC such that

‖xi+1− x′‖ ≤ ‖xi − x′‖ − C.

One can takeC = r 2/4R, so (ii) follows.

Proof of Theorem2.1. If X is a Delone set of finite type, then (2.2) certainly holds.To show the converse, suppose that (2.2) holds. Givenx, x′ ∈ X, by Lemma 2.2 there

exists a chain of points (2.4) inX connectingx to x′ that satisfies conditions (i) and (ii).These conditions together imply that(X− X)∩ B(0; T) is finite for any fixedT . To seethis, note that the chain (2.4) connecting two pointsx, x′ ∈ X with ‖x− x′‖ ≤ T has atmostCT steps by (ii), and there are at mostDX(2R) choices for each stepxi+1 − xi inthe chain, by (i). ThusDX(T) = #{(X − X)∩ B(0, T)} is finite for all T , henceX is offinite type.

Finally, the existence for allx′ ∈ X of the chain (2.4) satisfying (i) and (ii) shows that

[X] := Z[X] ⊆ Z[x0 ∪ {x : x ∈ (X − X) ∩ B(0;2R)}].

Thus (2.3) follows.

The finiteness criterion of Theorem 2.1 carries over to finiteness of local neighbor-hoods of the setX under translations.

Definition 2.2. (i) The T -neighborhoodof a pointx ∈ X is the set

PX(x, T) := X ∩ B(x; T) = {x′ ∈ X : ‖x′ − x‖ ≤ T}. (2.5)

A T -patchof X is aT-neighborhood ofX regarded as movable under translations.(ii) A centered T -patchP is aT-neighborhood translated to the origin, i.e.,

P := PX(x; T)− x. (2.6)

Definition 2.3. (i) The T-atlasAX(T) of a Delone setX is the set of all centeredT-patches ofX, i.e.,

AX(T) := {X ∩ B(0; T)− x : x ∈ X}. (2.7)

(ii) The atlas counting function N∗X(T) counts the number of different centeredT-patches, i.e.,

N∗X(T) := |AX(T)|, for T > 0. (2.8)

170 J. C. Lagarias

Corollary 2.1a. Let X be a Delone set with relative denseness constant R. Then X is aDelone set of finite type if and only if it has finitely many different centered2R-patches,i.e., N∗X(2R) <∞.

Proof. This follows from Theorem 2.1, since ifX has finitely many different 2R-patches thenDX(2R) is finite, while if DX(2R) is finite, then there are finitely manychoices for points within distance 2R of a fixed pointx.

Delone sets of finite type arise from tilings ofRn by translates of a finite nonemptyset of prototiles. We suppose that each prototile is simply connected, is the closure ofits interior, and has a boundary of measure zero. To any such tilingT we can associatea Delone set (in many different ways) by marking a finite set of points in each prototile,and takingX to be the set of marked points in the tiling. We say that such a tilingT hasafinite number of local patchesif it contains only finitely many translation-inequivalentarrangements of neighbors of each prototile. We have:

Corollary 2.1b. Let T be a tiling ofRn by translations of a finite set of prototiles.Suppose that each prototile is decorated by a finite nonempty set of points, and letX = X(T ) be the point set associated toT by these decorations. If T has a finitenumber of local patches, then X(T ) is a Delone set of finite type, and conversely.

Proof. This follows from Theorem 2.1 because there are only finitely many translation-inequivalent ways to tile neighborhoods of diameter 2R.

Baake et al. [3] (see also [1] and [2]) introduced a notion of “local derivability”between tilings, which has the following analogue for Delone sets.

Definition 2.4. (i) Let X andY both be Delone sets which are relatively dense withconstantR. We say thatY is locally derivablefrom X if there is a constantR′ suchthat for eachx ∈ X the set of points ofY ∩ B(x;2R) is completely determined by thecenteredR′-patch ofX aroundx. That is, the point setPx(Y;2R) = (Y ∩ B(x; R))− xis a function of the centered patchPx(X, R′) = (X ∩ B(x; R′))− x.

(ii) We say thatX andY aremutually locally derivableif each is locally derivablefrom the other.

Corollary 2.1c. If X is a Delone set of finite type and Y is a Delone set locally derivablefrom X, then Y is a Delone set of finite type.

Proof. By hypothesisDX(R′) is finite. Since the translation classes ofR′-patches of Xdetermine the 2R-patches ofY, we haveDY(2R) is finite, and Theorem 2.1 applies.

Our next object is to derive several equivalent criteria for a setX to be a Delone setof finite type. To formulate these, we make some definitions.

Geometric Models for Quasicrystals, I 171

Definition 2.5. A Delone setX has thelocally finite atlas property under translationsif N∗X(T) is finite for all T > 0.

Definition 2.6. (i) Given a discrete setX in Rn and a pointx ∈ X theVoronoi domainVX(x) atx is the set of all points inRn as close tox as to any other point ofX, namely,

VX(x) = {y : ‖y− x‖ ≤ ‖y− x′‖, all x′ ∈ X}.A marked Voronoi domainis a Voronoi domain together with a specific point in it viewedas a marked point.

(ii) The set of Voronoi domains of a discrete setX form a tiling of Rn called theVoronoi tesselationof Rn induced byX. A marked Voronoi tesselationconsists of theVoronoi tesselation induced by a discrete setX with the points ofX marked.

If X is a Delone set, then each Voronoi domainVX(x) is a bounded convex polytopecontained inB(x;2R). (For a general discrete setX the setVX(x) is a convex poly-tope, possibly unbounded.) The setX cannot always be uniquely reconstructed from itsVoronoi tesselation. It is uniquely determined by the induced Voronoi tesselation plusa single Voronoi domain marked with a point ofX; this determines the marking in allother Voronoi domains by the points ofX.

Definition 2.7. (i) Given a Delone setX in Rn a finite subsetY of X determines anempty sphereif all points ofY lie on a sphere inRn, there are no points ofX in the interiorof this sphere, and the convex hull ofY has nonempty interior. We call the convex hullof all points ofX lying on this sphere theDelaunay domaindetermined byY in X. It isa convex polytope.

(ii) The set of Delaunay domains of a Delone setX give a tiling ofRn called theDelaunay tesselationof Rn induced byX.

The Delaunay tesselation was introduced by Delone [9] in 1934. He showed thatfor a Delone setX in “general position” all Delaunay domains are simplices; thus theDelaunay tesselation is often called theDelaunay triangulationin the computationalgeometry literature,4 see p. 49 of [58]. A proof that Delaunay domains give a facet-to-facet tiling ofRn can be found in Fortune [15], and in the “generic case” in Chapter 8of [68]. The setX is uniquely reconstructible from its associated Delaunay tesselation,becauseX is the set of all vertices of all Delaunay domains. For general references onVoronoi and Delaunay tesselations, see [14] and [15].

Theorem 2.2. For a Delone set X inRn, the following properties are equivalent:

(i) X is a Delone set of finite type.(ii) X has the locally finite atlas property under translations.

4 The spellingDelaunay triangulationis standard in the computational geometry literature. Here “generalposition” means that there do not existn+ 2 points ofX lying on any sphere of diameter at most 2R. Rogers[68, Chapter 8] requires instead that the setX have non+2 equidistant points; in this case a Delaunay domainas we have defined it can be triangulated in a canonical way to obtain his “Delaunay triangulation.”

172 J. C. Lagarias

(iii) The marked Voronoi tesselation ofRn induced by X has finitely many translation-inequivalent marked Voronoi domains.

(iv) The Delaunay tesselation ofRn induced by X has finitely many translation-inequivalent Delaunay domains.

(v) X is finitely generated and any address mapϕ: [X] → Zs has

‖ϕ(x)− ϕ(x′)‖ ≤ C0‖x− x′‖, all x, x′ ∈ X, (2.9)

for some constant C0 depending onϕ.

Proof. (i)⇒ (ii) Since(X − X) ∩ B(0; T) is finite, and forx ∈ X all points in

P(x; T) := (X − x) ∩ B(0; T) (2.10)

lie in this set, there are only finitely many choices forP(x; T). In fact

N∗X(T) ≤ 2DX(T). (2.11)

(ii) ⇒ (iii) A marked Voronoi domainVX(x) is completely determined by the patch

P(x;2R) := (X − x) ∩ B(0;2R).

Any point y with ‖x − y‖ > 2R is within distance 2R of some pointx′ ∈ X, hencecannot be inVX(x). The locally finite atlas property says that there are onlyN∗X(2R) <∞such patches, hence there are at mostN∗X(2R) possible translation-inequivalent Voronoidomains.

(iii) ⇒ (i) The Delone property ofX guarantees that the ballB(x; 12r ) is contained

in the Voronoi domainVX(x). The facets of each marked Voronoi domain determine themarking in each neighboring Voronoi domain, namely, drop a perpendicular fromx tothe hyperplane determined by the facet and extend it an equal distance to the other side.By hypothesis there are only a finite number of choices for each neighboring domain,up to translation. Continue adding marked Voronoi domains in this way to fill up theentire ball of radius 2R aroundx. One has to add at most 2R/r concentric layers ofVoronoi domains aroundx to do this. There are only finitely many possible extensions,and this locates all possible locations for elements ofX within distance 2R of X. Sincethis construction is translation-invariant, we conclude that(X− X)∩ B(0;2R) is finite.Now (i) follows by Theorem 2.1.

(i)⇒ (iv) A Delaunay domain including a given pointx ∈ X is completely determinedby all points within distance 2R of X, since the largest empty sphere touchingX has di-ameter at most 2R. By hypothesis (i) there are only finitely many translation-inequivalentarrangements of such points.

(iv) ⇒ (i) The Delaunay tesselation ofX uniquely determinesX. Given any twoelements ofX within a distance 2R, there is a path of edges in the Delaunay tesselationconnecting them which remains in a ball of radius 4R about one of the points. Since aDelaunay tesselation meets vertex-to-vertex, hypothesis (iv) implies that there are onlyfinitely many ways to tile a ball of radius 4R. ThusDX(2R) is finite, and Theorem 2.1implies thatX is a Delone set of finite type.

Geometric Models for Quasicrystals, I 173

(i)⇒ (v) A Delone setX of finite type is finitely generated by Theorem 2.1. To showthe other part, suppose thatϕ: [X] → Zs is an address map and set

C2 := max{‖ϕ(y)‖ : y ∈ (X − X) ∩ B(0;2R)}.Givenx, x′ ∈ X by Lemma 2.2 there exists a chain (2.4) at points ofX connectingx to x′

that satisfies‖xi+1− xi ‖ ≤ 2R and the number of points is at most(4R/r 2)‖xi+1− xi ‖.Using the linearity ofϕ on [X], we have

‖ϕ(x)− ϕ(x′)‖ ≤m∑

i=1

‖ϕ(xi+1)− ϕ(xi )‖

=m∑

i=1

‖ϕ(xi+1− xi )‖

≤ C2m≤ 4C2R

r 2‖x− x′‖,

which proves (iv) withC0 = 4C2R/r 2.(v)⇒ (i) If x, x′ ∈ X satisfy‖x′ − x‖ ≤ 2R, then by hypothesis

‖ϕ(x− x′)‖ = ‖ϕ(x)− ϕ(x′)‖ ≤ 2C R.

Addresses all lie inZs, hence there are only finitely many choices forϕ(x− x′), hencefor x− x′, sinceϕ: X → Zs is one-to-one. Thus(X − X) ∩ B(0;2R) is finite, soX isof finite type by Theorem 2.1.

Theorem 2.2 yields an upper bound on the growth of the distance counting functionDT (X).

Corollary 2.2. Let X be a Delone set of finite type, of rank s. There exists a positiveconstant C1 depending on X such that

DX(T) ≤ C1Ts, all T ≥ 1. (2.12)

Proof. Suppose thatx, x′ ∈ X with ‖x − x′‖ ≤ T . The pointx − x′ is determined byits addressϕ(x− x′), and, by (iv) of Theorem 2.2, there is a constantC0 such that

‖ϕ(x− x′)‖ ≤ C0‖x− x′‖ = C0T.

The set of elements inZs in a ball of radiusT is at mostCTs for all T ≥ 1, and (2.12)follows.

We next obtain an upper bound on the atlas counting functionN∗X(T) of a Delone setof finite type.

Definition 2.8. Theconfigurational entropy Hc(X) of a Delone setX in Rn is givenby

Hc(X) := lim supT→∞

log N∗X(T)Tn

.

174 J. C. Lagarias

We prove:

Theorem 2.3. If X is a Delone set of finite type inRn, there is a constant C2 dependingon X such that

N∗X(T) ≤ exp(C2Tn), all T ≥ 1. (2.13)

That is, X has finite configurational entropy.

Remark. The factorTn in the bound (2.13) cannot be improved, see Example 3.1 inSection 3.

Proof. We may build up all possible patches of radiusT out of cubical patchesof theform X ∩ C(x;8R) for x ∈ X, whereC(x; ρ) denotes a cube of side 2ρ centered atx. SinceX ∩ C(x;8R) ⊆ X ∩ B(x;8√nR), the numberC′ of translation-inequivalentcubical patchesX ∩ C(x;4R) is at mostDX(8

√nR).

We bound the number of different cubical patches of side 2T centered at0, all ofthose subpatches of side 2R match those in the listL. To begin, there areC′ choices forthe patch centered at0. Now we proceed to add cubical patches in concentric layers. Thecenters of the new patches are chosen at already constructed points, within distanceRof the boundary, so the new cubical patch extends at least 3R outside. At thekth stagewe have covered a cube of side 8+6k R. We can arrange that all cube centers chosen areseparated by at least 2R, hence there are at mostO((T/R)n) cubical patches to choosein the process. Thus there are at most(C′)C

′′Tnpossible choices of cubical patches of

side 2T , which proves (2.13).

We conclude this section with an example showing that the requirement thatX − Xbe closed is necessary in the definition of a Delone set of finite type. The Delone set

X ={

n+ 1

3n2+ 3: n ∈ Z

}inR has the property thatX− X is discrete. However,X− X has 1 as a finite limit point,so X is not a Delone set of finite type; it is not even finitely generated.

3. Meyer Sets

Meyer [53] recently defined a mathematical notion which he called a “quasicrystal,”which is based on concepts in harmonic analysis that he introduced around 1970, see[51] and [52]. These sets are termedMeyer setsby Moody [54], [55], who unifies andextends the known characterizations of a Meyer sets. Moody [55] takes a harmonicanalysis criterion as the basic definition of a Meyer set (relatively dense harmoniousset), while Meyer [53] defines them to be those Delone setsX such thatX− X ⊆ X+ Ffor some finite setF . These properties are equivalent toX − X being a Delone set, see[38].

It is known that Meyer sets can also be characterized in terms of cut-and-project sets.

Geometric Models for Quasicrystals, I 175

Definition 3.1. Let3 be a full rank lattice inRd = Rn+m = Rn ×Rm, and letπ‖ andπ⊥ be orthogonal projections onto the factorsRn andRm, respectively. AwindowÄ isa bounded open subset ofRm, and thestrip S(Ä) in Rd associated to the windowÄ is

S(Ä) := Rn ×Ä = {w ∈ Rd : π⊥(w) ∈ Ä}.

Thecut-and-project set X(3,Ä) associated to the data(3,Ä) is

X(3,Ä) = π‖(3 ∩ S(Ä)). (3.1)

We call d the dimensionof the data(3,Ä). A given cut-and-project setX maybe constructed in many ways, using different pairs(3,Ä) and (3′, Ä′) of differentdimensions. (Some authors allow the windowÄ to be a closed set, but we do not.)

An alternate definition of cut-and-project sets, which is used in [24] and [59], appearsslightly different but gives the same sets. It takes a fixed latticeZd in Rd, and projectsit onto an arbitrary pair(E‖, E⊥) of orthogonal subspaces of dimensionsn and m,respectively, using orthogonal projectionsπ‖ andπ⊥, respectively. It identifiesE‖ withRn by a linear map, and takes a window setÄ in E⊥.

Definition 3.2. A cut-and-project set isnondegenerateif π‖: Rd → Rn is one-to-oneon3, i.e., if

3 ∩ ({0} × Rm) = {(0,0)}.It is irreducible if it is nondegenerate andπ⊥(3) is dense inRm.

An irreducible cut-and-project set coincides with the concept of amodel setof Meyer[52]. Meyer proved that a Delone setX is a Meyer set if and only if there is an irreduciblecut-and-project setX′ and a finite setF such that

X ⊆ X′ + F ′, (3.2)

see [52], [53], and [55].

Meyer sets are Delone sets of finite type, hence are finitely generated. We characterizewhich finitely generated Delone sets are Meyer sets in terms of their address maps, usingthe following concept due to Meyer.

Definition 3.3. A mapψ : [X] → Rd is analmost linear mappingon X if there existsa linear mapLψ : Rn→ Rd such that

‖ψ(x)− Lψ(x)‖ < C, all x ∈ X. (3.3)

Condition (3.3) applies only tox ∈ X and not to general elements of [X].

The following result gives several characterizations of Meyer sets. These character-izations are mainly due to Meyer (see also [55]); the equivalence of (iv) and (v) seemsto be new.

176 J. C. Lagarias

Theorem 3.1. The following properties of a discrete set X inRn are equivalent:

(i) X is a Meyer set. That is, X − X is a Delone set.(ii) X is a Delone set and there is a finite set F such that

X − X ⊆ X + F.

(iii) X is a finitely generated Delone set and every homomorphismψ : [X] → Rd

for some d≥ 1 is an almost linear mapping on X.(iv) X is a finitely generated Delone set and the address mapϕ: [X] → Zs is an

almost linear mapping on X, i.e., there is a linear mapL: Rn → Rs and aconstant C such that

‖ϕ(x)− L(x)‖ ≤ C, all x ∈ X. (3.4)

(v) X is a finitely generated Delone set and there exists a nondegenerate cut-and-project set X′ of dimension at most rank(X) such that

X ⊆ X′. (3.5)

Remark. The cut-and-project setX′ containingX that appears in (v) is not necessarilyirreducible.

Proof. (i)⇔ (ii) This is proved in [38].(ii) ⇒ (iii) This result is due to Meyer and a proof can be found in Section 8 of [55].

We mention some important points. The mapL: Rn → Rs can be constructed as an“ideal address map,” as follows. For eachy ∈ Rn define

L(y) := limk→∞

ϕ(xk)

2k, (3.6)

wherexk ∈ X satisfies‖xk − 2ky‖ ≤ R. Using the Meyer set property one proves thatthis limit exists and is unique, and satisfies

L(cy+ y′) = cL(y)+ L(y′), y, y′ ∈ Rn,

hence is linear. We callL an ideal address mapbecause it satisfies the identity

y =s∑

j=1

L(y)j vj , all y ∈ Rn. (3.7)

This identity is derived from (3.6) using the address identity

x =s∑

j=1

ϕ(x)j vj , all x ∈ [X].

(iii) ⇒ (iv) Immediate.(iv)⇒ (ii) Given x, x′ ∈ X, hypothesis (iv) gives

‖L(x− x′)− ϕ(x− x′)‖ ≤ C. (3.8)

Geometric Models for Quasicrystals, I 177

SinceX is a Delone set, there is some pointx′′ ∈ X with

‖x′′ − (x+ x′)‖ ≤ R,

and by hypothesis

‖L(x′′)− ϕ(x′′)‖ ≤ C. (3.9)

Now

‖L(x′′ − x+ x′)‖ ≤ C′‖x′′ − (x+ x′)‖ ≤ C′R, (3.10)

whereC′ = ‖L‖. The triangle inequality applied to (3.8)–(3.10) gives

‖ϕ(x′′ − (x− x′))‖ = ‖ϕ(x′′)− ϕ(x− x′)‖ ≤ 2C + C′R, (3.11)

hence there are a finite number of choices for addresses inZs satisfying (3.11). Since theaddress map is one-to-one on [X], there are a finite number ofy = x′′ − (x− x′) ∈ [X]satisfying (3.11), and we call this setF . We have shown that

X − X ⊆ X + F,

which proves (ii).(iv)⇒ (v) We use the almost linear mappingL: Rn→ Rs associated to the address

map by (3.7) to construct a suitable nondegenerate cut-and-project set. Sets = n+m,som≥ 0, and let the vector space

V := {L(x) : x ∈ Rn}

be the image of the mapL. We first show that dim(V) = n. Defineψ : Rs→ Rn by

ψ(u1, . . . ,us) =s∑

j=1

uj vj

and observe that (3.7) gives

ψ ◦ L(y) = y all y ∈ Rn.

Thusψ(V) = Rn hence dim(V) = n. Now set

W := ker(ψ) ={(`1, . . . , `s) :

s∑j=1

`j vj = 0

}.

Now dim(W) = s− n = m. Choose an orthonormal basis ofW,

`(i ) = (`i 1, . . . , `is), 1≤ i ≤ m,

and set

wj := L(vj )− ej , 1≤ j ≤ s,

178 J. C. Lagarias

whereej = (0, . . . ,1, . . . ,0) is the j th coordinate vector. Now (3.7) implies thatwj ∈W, hence we may define coefficients{αj i } by

wj =m∑

i=1

αj i `(i ), 1≤ j ≤ s. (3.12)

Let3 = Z[v∗1, . . . , v∗s] be the additive subgroup ofRs spanned by

v∗j := (vj , αj 1, αj 2, . . . , αjm), 1≤ j ≤ s.

We claim that3 is a full rank lattice inRd. To show this, it suffices to prove that thevectorsv∗j areR-linearly independent. So suppose that

s∑j=1

uj v∗j = 0.

The firstn coordinates give

s∑j=1

uj vj = 0,

henceu = (u1, . . . ,ud) ∈ W. Now the lastm coordinates give

s∑j=1

ujαj i = 0, 1≤ i ≤ m.

Thuss∑

j=1

uj wj =s∑

j=1

uj

(m∑

i=1

αj i `(i )

)=

m∑i=1

`(i )

(s∑

j=1

ujαj i

)= 0,

and the definition ofwj yields

s∑j=1

uj L(vj )− (u1,u2, . . . ,us) = 0. (3.13)

Since each coordinate is linear,

(u1, . . . ,us) = L

(s∑

j=1

uj vj

)= L(0) = 0.

Thus [v∗1, . . . , v∗s] are linearly independent, and3 is a full rank lattice inRs.

The projectionπ‖: Rs→ Rn sends

π‖(wj ) = vj , 1≤ j ≤ s,

hence it is one-to-one on3 since [X] = Z[v1, . . . , vs] is a free module.

Geometric Models for Quasicrystals, I 179

Now suppose thatx ∈ [X] and its address is

x =s∑

j=1

ϕ(x)j vj , with ϕ(x)j ∈ Z.

Then

x∗ :=s∑

j=1

ϕ(x)j wj

hasx∗ ∈ 3, andπ‖(x∗) = x. Now set

π⊥(x∗) =s∑

j=1

ϕ(x)j (αj 1, αj 2, . . . , αjm) = (z1, . . . , zm).

Since the vectors(i ) are orthonormal, we have

‖π⊥(x∗)‖ =(

m∑j=1

z2i

)1/2

=∥∥∥∥∥ m∑

i=1

zi `(i )

∥∥∥∥∥=∥∥∥∥∥ m∑

i=1

s∑j=1

ϕ(x)jαj i `(i )

∥∥∥∥∥=∥∥∥∥∥ s∑

j=1

ϕ(x)j wj

∥∥∥∥∥= ‖ϕ(x)− L(x)‖,

where (3.12) and (3.13) were used at the last step. The almost linear property (3.4) thenyields

‖π⊥(x∗)‖ ≤ C.

It therefore suffices to take the window set

Ä = {z ∈ Rm : ‖z‖ < C + 1}to obtain a nondegenerate cut-and-project setX(3,Ä) of dimensions with

X ⊆ X(3,Ä),

which is (v).(v)⇒ (iv) This is a result of Meyer, for which see Theorem 9.1 of [55]. Alternatively

one can directly reconstruct the mapL(x) from X′, using

L(x) := (π‖)−1(x) ∩3 for x ∈ X.

This is well defined sinceX′ is nondegenerate. NowL extends to a linear map on [X],and (3.4) holds by the cut-and-project property ofX′.

180 J. C. Lagarias

The following result of Meyer follows directly from property (v) above.

Corollary 3.1a. If X is a Meyer set and Y is a Delone set with Y⊆ X, then Y is aMeyer set.

We obtain a similar result for sets locally derivable from Meyer sets.

Corollary 3.1b. If X is a Meyer set and Y is a Delone set locally derivable from X,then Y is a Meyer set.

Proof. We claim thatY ⊆ X + F ′ for some finite setF ′. Indeed there are only finitelymany differentR′-patches ofX, and the elements ofY can be viewed as obtained byindividually shifting the center of eachR′-patch ofX, in a finite number of differentways, so the claim follows. SinceX is a Meyer set, the setX′ = X + F ′ is a Meyer setbecause it satisfies property (ii) above:

X′ − X′ ⊆ X′ + (F + F ′ − F ′). (3.14)

Now Y is a Meyer set by Corollary 3.1a.

We also apply Corollary 3.1a in the following example.

Example 3.1. There exists a Meyer setX ⊆ Zn in Rn such that

N∗X(T) ≥ exp(C3Tn), all T ≥ 1, (3.15)

with C3 = 12 log 2.

We will arrange that 2Zn ⊆ X ⊆ Zn, which guarantees thatX is a Meyer setby Corollary 3.1a. We use a simple random construction: pick each point inZn\2Zn

independently to be inX with probability 1/2. Now consider a box of side 2nT centeredat a point in 2Zn. Every possible assignment of cells in the box not in the lattice 2Zn

occurs with positive probability, hence for any fixed assignment the possibility that nosuch configuration occurs inX is zero. The union over all such exceptional events, forT = 1,2,3, . . . , then has probability zero, hence with probability one every possibleassignment occurs, for everyT ≥ 1. Thus, with probability one,

N∗X(2nT) ≥ exp((log 2)(2n − 1)Tn), for all T ≥ 1,

as required.There are much stronger examples known than the one above. A Delone setX of finite

type isminimal if it has therepetitivity property: For eachT > 0 there is a finite valueMX(T) such that every ball of radiusMX(T) contains a copy of each patch ofX of radiusT . Furstenberg [16, Theorem III.2] constructs a symbolic sequence giving a minimalset with positive entropy. By using intervals of different integer lengths to represent thesymbols, we obtain a one-dimensional Meyer set which is minimal and satisfies (3.15)for someC3. Pleasants [61] constructs such examples in arbitrary dimensions. Minimalsets are important because they are an analogue of “ground state” configurations, see[64], and the introduction to part II.

Geometric Models for Quasicrystals, I 181

4. Inflation Symmetries

We recall the basic definition.

Definition 4.1. A Delone setX in Rn has aninflation symmetryby the real numberη > 1 if ηX ⊆ X.

If 0 6∈ X, then X′ = X ∪ {0} is also a Delone set with an inflation symmetry byη, so without loss of generality we may assume that0 ∈ X. An inflation symmetry isnot a complete symmetry of a Delone setX, because the setX\ηX contains infinitelymany points. Inflation similarities occur in Delone sets constructed as control points ofself-similar tilings, see [32] and [74].

A (complex) numberη is analgebraic integerif f (η) = 0 for some nonzero monicpolynomial f (u) ∈ Z[u], i.e.,

f (u) = un +n−1∑j=0

aj uj .

The degreeof η is the minimal degree of any nonzero polynomialf (u) ∈ Z[u] withf (η) = 0. Thealgebraic conjugatesof η are the other roots of the minimal degree monicpolynomial inZ[u] which η satisfies.

Several classes of real algebraic integers appear in studying inflation symmetries.

Definition 4.2. Let η be a real algebraic integer withη > 1.

(i) η is aPisot numberor Pisot–Vijayaraghavan numberif all algebraic conjugates|η′| < 1.

(ii) η is aSalem numberif all algebraic conjugates|η′| ≤ 1 and at least one|η′| = 1.(iii) η is aPerron numberif all algebraic conjugates|η′| < η.(iv) η is aLind numberif all algebraic conjugates|η′| ≤ η and at least one|η′| = η.

Pisot and Salem numbers have been extensively studied, see [4]. Perron numbers wereintroduced and studied by Lind [46]–[48] and Lind numbers are introduced here.

Theorem 4.1. Let X be a Delone set inRn such thatηX ⊆ X for a real numberη > 1.

(i) If X is finitely generated, thenη is an algebraic integer. If X has rank s, then thedegree ofη divides s.

(ii) If X is a Delone set of finite type, thenη is an algebraic integer with all algebraicconjugates|η′| ≤ η. That is, η is a Perron number or a Lind number.

(iii) If X is a Meyer set, thenη is an algebraic integer with all algebraic conjugates|η′| ≤ 1. That is, η is a Pisot number or a Salem number.

Proof. (i) Let 0 ∈ X and suppose thatX has ranks, so that

[X] = Z[X] = Z[v1, . . . , vs] (4.1)

182 J. C. Lagarias

for suitable vectors{vi : 1 ≤ i ≤ s} independent overQ. Now ηX ⊆ X implies thatη[X] ⊆ [X], hence

ηV = MV , with V =

v1...

vs

, (4.2)

in whichM is ans× s integral matrix andV is ans× n real matrix. Now

f (x) = det(λI −M) ∈ Z[λ]

is a monic polynomial, andf (η) = 0 sinceηI−M annihilates ann-dimensional subspaceof Rs. Thusη is an algebraic integer, of degree at mosts.

To show thatd = degree(η) dividess, we use the rational canonical form ofM . Letf (λ) ∈ Z[λ] be the minimal degree monic polynomial thatη satisfies, and write

f (λ) = λd +d∑

j=1

ajλd− j .

Now (4.2) givesη j V = M j V for 1≤ j ≤ d, hence

f (M)V =(

Md +d∑

j=1

aj Md− j

)V = 0. (4.3)

Each row of f (M) is an integer vector, so (4.2) gives

s∑j=1

f (M)i j vj = 0, 1≤ i ≤ s. (4.4)

The vectorsvj are independent overQ, so it follows that allf (M)i j = 0, hencef (M) =0. Thus the minimal polynomial ofM divides f (λ), and sincef (λ) is irreducible overQ, it must equalf (λ). It follows that the rational canonical form ofM (overQ) can onlybe a block diagonal matrix with all blocks equal to thed × d companion matrixC f off , see p. 199 of [27]. Thusd dividess.

(ii) Since X is a Delone set of finite type, by Theorem 2.1 it is finitely generated,of rank s, say. Nowη is a real algebraic integer by (i). Letη be of degreed and setZ[η] := Z[1, η, . . . ηd−1], which is a ring.

We now argue by contradiction, and suppose thatη′ is an algebraic conjugate ofηwith |η′| > η > 1. We first observe that theZ-module [X] is actually aZ[η]-module,because ify ∈ [X], then so isηy, since we can expressy as a finite integral combinationof elements ofX, and if x ∈ X, thenηx ∈ X. Now [X] is not necessarily a freeZ[η]-module, but there exists a freeZ[η]-module [W] in Rn containing [X] as a submoduleof finite index. It can be obtained from the matrix convertingM to canonical form in (i)above. Nowd dividess and we have

[X] ⊆ [W] = Z[η][w1,w2, . . . ,ws/d],

Geometric Models for Quasicrystals, I 183

i.e., there is someZ-basis [v1, . . . , vs] of [W] with

vi+( j−1)s/d = ηi−1wj , 1≤ i ≤ d, 1≤ j ≤ s/d. (4.5)

We now define theZ[η′]-module

[W′] := Z[η′][w1,w2, . . . ,ws/d],

which we regard as a set of points inCn, viewingη′ ∈ C even ifη′ is real. We call thisspaceCn theshadow space, and we measure distances on it, using the Hermitian norm

‖z− z′‖2H :=n∑

i=1

|zi − z′i |2, where z= (z1, . . . , zn) ∈ Cn.

We define ashadow mapσ : [W] → [W′] by σ(wj ) = wj for 1 ≤ j ≤ s/d and extendit using the Galois map fromZ[η] to Z[η′], so that

σ(y1+ y2) = σ(y1)+ σ(y2),

σ (ηy1) = η′σ(y1)

for y1, y2 ∈ [X]. Let [X′] denote the image of [X] under the shadow map, and choose apointx ∈ X whose imageσ(x) 6= 0. We obtain a contradiction by measuring the lengthof the shadow image of the vectorsyk = ηkx − x in two different ways, ask → ∞.First,σ(yk) = (η′)kσ(x)− σ(x) hence

‖σ(yk)‖ = |(η′)k − 1| ‖σ(x)‖H

≥ c0|η′|k as k→∞. (4.6)

On the other hand, bothx, ηkx ∈ X and using the Delone property ofX we can find apath of elementsx0, x1, . . . , xm ∈ X from x0 = x to xm = Tkx, where

‖xk − xk−1‖ ≤ 3R, 1≤ k ≤ m, (4.7)

andm = (1/R)‖(ηk − 1)x‖. Indeed, draw a straight line connectingx andηkx in Rn,and by the Delone property there is a vectorxk ∈ X within distanceR of the point(1+ k(‖X‖/R))x, and the bound (4.6) follows from the triangle inequality. SinceX isof finite type, there are only finitely many vectors in

(X − X) ∩ B(0,3R),

hence

C := max{‖σ(y)‖H : y ∈ (X − X) ∩ B(0;3R)} (4.8)

is finite. It follows from (4.7) that

‖σ(xk − xk−1)‖H ≤ C. (4.9)

Now the linearity ofσ on [X] gives

σ(yk) =m∑

i=1

σ(xk)− σ(xk−1),

184 J. C. Lagarias

hence (4.9) gives

‖σ(yk)‖H ≤m∑

i=1

‖σ(xk)− σ(xk−1)‖H

≤ mC= 1

R(ηk − 1)‖x‖C

≤ C1ηk as k→∞.

If |η′| > η, this contradicts (4.6) fork large enough, so (ii) follows.(iii) This is a result of Meyer [53, Theorem 6]. Here we present an alternative proof.

SinceX is a Meyer set, the address mapϕ: [X] → Rs is an almost linear map. LetL: Rn→ Rs be the (unique) linear map such that

|ϕ(x)− L(x)| ≤ C, all x ∈ X. (4.10)

Part (i) of this proof showed that there is an integer matrixM such that

ϕ(ηx) = Mϕ(x), all x ∈ [X], (4.11)

and that the minimal polynomial ofM is the monic irreducible polynomialf (λ) ∈ Z[λ]thatη satisfies. Write

f (λ) =d∏

i=1

(λ− ηi ) = λd +d∑

j=1

ajλd− j ,

with η = η1. Part (i) also showed thatM is diagonalizable overC, and has a direct sumdecomposition into eigenspaces

Cn = V1+ V2+ · · · + Vd, (4.12)

in which Vj is the eigenspace for the eigenvalueηj . We also have

Mv = ηj v for all v ∈ Vj , 1≤ j ≤ d,

and dim(Vj ) = s/d. The imageϕ([X]) spansRs, so there existsx ∈ X with ϕ(x) 6= 0.Now ϕ(x) uniquely decomposes into eigenvectors using (4.12), i.e.,

ϕ(x) =d∑

j=1

vj , with each vj ∈ Vj .

Sinceϕ(x) ∈ Zs\{0}, everyvj 6= 0, because the Galois group off (λ) permutes all thevj transitively.

Iterating (4.11) gives

M kϕ(x) =d∑

j=1

(ηj )kvj , all k ≥ 1. (4.13)

Geometric Models for Quasicrystals, I 185

On the other hand, (4.10) gives

|L(ηkx)− ϕ(ηkx)| = |ηk L(x)−M kϕ(x)| ≤ C.

Combining this bound with (4.13) yields∥∥∥∥∥ηk(L(x)− v1)−d∑

j=2

(ηj )kvj

∥∥∥∥∥ ≤ C, all k ≥ 0. (4.14)

We claim that this implies thatL(x) = v1 and that|ηj | ≤ 1 for all j ≥ 2. To show this,consider thei th coordinate vector. Then (4.14) gives|h(i )k | ≤ C, where

h(i )k =d∑

j=1

vj i (ηj )k, (4.15)

wherevj i is the i th entry ofL(x) − v1 for j = 1 and of−vj for j ≥ 2. This relationyields the identity

∞∑k=0

h(i )k zk =d∑

j=1

vj i

1− ηj z, 1≤ i ≤ s,

where both sides converge as functions of the complex variablez for sufficiently small|z|. The relations|h(i )k | ≤ C imply that the power series on the left converges in the unitdisk |z| < 1. Since allηj are distinct we must have allvj i = 0 for any|ηj | > 1. Thusv1i = 0 for 1≤ i ≤ s, henceL(x) = v1. Since eachvj 6= 0 for j ≥ 2, for each suchjthere exists somevj i 6= 0, hence we infer that|ηj | ≤ 1 for j ≥ 2.

Remarks. Each of (i)–(iii) is apparently sharp, in the sense that given any algebraicintegerη one can construct a one-dimensional Delone setX ⊆ R of the appropriatetype havingηX ⊆ X. For (iii) this was shown by Meyer. For (i) it is quite easy to givea recursive construction ofX ⊆ R, with [X] = Z[η], such that 0∈ X, X = −X,andηX ⊆ X. To do this, construct points in [1, η), then recursively construct points in[ηk, ηk+1) by first including all pointsηy for y ∈ [ηk−1, ηk), and adding some new pointsas necessary to preserve the Delone property. For (ii) this construction can apparentlybe carried out but requires ingenuity in choosing the correct new points to add; I haveverified it forη = √2, which is a Lind number. The proof of Theorem 4.1 above worksmore generally for real numbersη such that|η| > 1, with the condition in (ii) modifiedto require that all algebraic conjugates|η′| ≤ |η|. R. Kenyon (private communication)observes that the Delone set of finite typeX := {−n

√2 : n = 1,2, . . .} ∪ {n : n =

0,1,2, . . .} has an inflation by−√2.There are results in the literature for self-similar tilings which are analogous to The-

orem 4.1. Kenyon [33] proves a result analogous to (i) for tilings. The results for self-similar tilings (see [78], [32], and [34]) analogous to cases (ii) and (iii) differ slightly inthat they rule out inflation symmetry factorsη that have an algebraic conjugateη′ whichsatisfies the equality|η′| = η in (ii) or |η′| = 1 in (iii). That is, they rule out Lind numbersand Salem numbers in (ii) and (iii), respectively. R. Kenyon (private communication)

186 J. C. Lagarias

observes that if to Theorem 4.1 one adds the extra hypothesis that the Delone setX isrepetitive (i.e., minimal), then Lind numbers can be ruled out in (ii), and Salem numberscan be ruled out in (iii). The repetitivity property holds for self-affine tilings with theunique composition property, see [75].

5. One-Dimensional Delone Sets of Finite Type and Symbolic Dynamics

In this section we contrast properties of general Delone sets of finite type and Meyer setsby studying various one-dimensional examples. In particular, we consider diffractionproperties of such sets.

One-dimensional Delone sets of finite typeX are in one-to-one correspondence withtilings of the line by intervals of finitely many different lengths. IfX = {xn : −∞ < n <∞}with the points ofX numbered in increasing order, then we may regardX as tiling theline with intervalsIn = [xn, xn+1]. Theorem 2.1 implies that these intervals have finitelymany different lengths, call themα0, α1, . . . , αm. It also implies the converse assertionthat any such tiling gives a Delone setX of finite type. Such a tiling is determined bythe location of the pointx0 together with the symbolic expression

SX = (. . . , s−1, s0, s1, . . .) ∈ {0,1, . . . ,m}Z, (5.1)

in which si is a symbol indicating thetypeof the i th interval, and typej denotes aninterval of lengthαj . Since we only consider setsX up to translation, we may supposethatx0 = 0, in which case the symbol sequenceSX and the assignment of lengthsαj tosymbol typej completely describeX.

We now specialize to the case in whichX has intervals of only two lengths, 1 andα, in which case a symbol sequenceS ∈ {0,1}Z and the lengthα completely specify aDelone setX = Xα(S) with x0 = 0. We will show that conditions for the setsXα(S) tohave a unique autocorrelation measureγ and a unique diffraction measureγ depend onthe symbolic addressS but not onα, while in general the property ofX being a Meyerset depends onboth Sandα.

Definition 5.1. A symbol sequenceS∈ {0,1}Z isalmost linearif there is a real numberβ such that the partial sumsSn+1 := s0+s1+· · ·+sn andS−n := s−1+s−2+· · ·+s−n,for n ≥ 0, satisfy

|Sn − nβ| ≤ C, all n ∈ Z, (5.2)

for some constantC.

The following result characterizes whenXα(S) is a Meyer set.

Theorem 5.1. Let S∈ {0,1}Z be a symbol sequence. If S is almost linear, then Xα(S)is a Meyer set for all positiveα. If S is not almost linear, then Xα = Xα(S) is a Deloneset of finite type and satisfies:

(i) If α is rational, then rank(Xα) = 1 and Xα is a Meyer set.(ii) If α is irrational, then rank(Xα) = 2 and Xα is not a Meyer set.

Geometric Models for Quasicrystals, I 187

Proof. If α ∈ Q, thenXα(S) has rank 1 for anyS⊆ {0,1}Z. If α has denominatorm,thenXα(S) ⊆ (1/m)Z, henceSα(S) is a Meyer set by Corollary 3.1a.

If α 6∈ Q, then if S ∈ {0,1}Z is not 0∞ or 1∞, then Xα(S) has rank 2. Nowxn =(n−Sn)+Snα, so the address mapϕ: [X] → Z2 obtained from the basisv1 = 1, v2 = αis

ϕ(xn) = (n− Sn, Sn), all n ∈ Z.By Theorem 3.1Xα is a Meyer set if and only if there exists a linear functionL: R→ R2,sayL(x) = (β1x, β2x) such that

‖ϕ(xn)− L(xn)‖ ≤ C, all n ∈ Z,for some constantC. This gives

‖((1−β1)n+(β1α−β1−1)Sn, β2n+(β2α−β2−1)Sn)‖ ≤ C, all n ∈ Z. (5.3)

Each coordinate is bounded separately, and at least one of them implies (5.2), so thatfor Xα to be a Meyer set,Smust be almost linear. Conversely, ifS is almost linear, then(5.3) holds for the choice

β1 = 1− β1− β + αβ and β2 = β

1− β + α β ,

so Xα(S) is a Meyer set. This includes the cases whenS is 0∞ and 1∞.

Results analogous to Theorem 5.1 hold for setsX derived from symbol sequencesdrawn fromanalphabetwithm letters, and tilingRwith intervalsof lengthα0, α1, α2, . . . ,

αm−1.For results concerning autocorrelation measures and diffraction intensities for one-

dimensional setsX, we rely on results of Hof [24], [26] and Solomyak [73]–[76]. Recallfrom [24] that anautocorrelation measureγ of a Delone setX in Rn is any limit pointin the vague topology of the sequence of measures

γ L := (2L)−n∑

x,x′∈X∩[−L ,L]d

δx−x′ , for L = 1,2,3, . . . .

We are primarily interested in setsX that have a unique autocorrelation measureγ . Inorder for a unique autocorrelation measureγ for Xα(S) to exist, the symbol sequenceSmust have limiting frequencies for the tiles of lengths 1 andα, i.e., there must exist avalued such that

limn→∞

1

nSn = lim

n→∞1

nS−n = d. (5.4)

We can easily construct Meyer setsX for which such limits do not exist. For example,takeX = X2(S) with sn = s−n and with thesn for n ≥ 0 being given by

S+ =∞∏

k=1

(00)2k(01)2

k(11)2

k. (5.5)

Here X2(S) ⊆ Z is a Delone set withR = 2, hence is a Meyer set, and (5.4) fails tohold.

188 J. C. Lagarias

Definition 5.2. (i) A Delone setX of finite type is said to haveuniform patch frequen-cies if for each patchP = (X − x) ∩ B(0; T) with x ∈ X, and eachε > 0 there is avalueL = L(P, ε) such that for any interval of lengthL on the lineX contains copiesof the patchP having a density betweend(P)− ε andd(P)+ ε. That is,

d(P)− ε ≤ #{y : y+ P ⊆ X andu ≤ y ≤ u+ L}#{x : x ∈ X andu ≤ x ≤ u+ L} ≤ d(P)+ ε. (5.6)

(ii) We say that a symbol sequenceS⊆ {0,1}Z hasuniform patch frequenciesif thisholds for the Delone setXα(S) obtained fromSby takingα = 2.

It is easy to see that the symbol sequenceS has uniform patch frequencies if andonly if the same is true for allXα(S), α > 0. Solomyak [76] observes that this propertyimplies that the tiling dynamical system attached to each suchXα(S) is uniquely ergodic,and that eachXα(S) has a well-defined autocorrelation measureγα in the sense of Hof[24], [26]. Furthermore,γα is a pure discrete measure supported onX − X, with

γα =∑

y∈X−X

f (y)δy, (5.7)

in which f (y) is the uniform limiting frequency of occurrence of the distancey in X.The Fourier transformγα of the autocorrelation measure is the diffraction measure ofXα(S), and is a positive measure.

Solomyak [74, Theorem 5.1] gives sufficient conditions under whichX has uniformlimiting patch frequencies. In particular, these conditions are satisfied forS arising asfixed points of irreducible substitutions, andX = Xα(S) have the properties above forall α > 0. SuchXα(S) are non-Meyer sets ifα is irrational and the largest eigenvalue ofthe substitution systemS is not a Pisot number.

Under the assumption thatS has uniform patch frequencies, each setXα(S) has awell-defineddiffraction measureγα. The occurrence of a nonzero discrete component inγα depends on bothSandα, as is shown by examples in [36]. The examples suggest thata two-letter irreducible substitution system can only have a nonzero discrete componentwhenXα(S) is a Meyer set. The associated expansion constant of the substitutionSis thennecessarily a Pisot number. Solomyak [73] has shown there exist multiletter substitutionrules with largest eigenvalue not a Pisot number, which have a discrete component in thespectrum of the associated substitution dynamical system. (See also [13].) At present itis not known whether there exist non-Meyer setsXα(S) with Shaving uniform limitingpatch frequencies, such that the diffraction measureγ contains some nontrivial discretespectrum.

Several authors have considered analogous problems for self-similar tilings and tilingdynamical systems. G¨ahler and Klitzing [18] assert that the set of control points of aself-similar tiling with inflation factorη can have a nontrivial discrete component in thediffraction measure only ifη is a Pisot number. Solomyak in [74, Theorem 8.1] and [75,Theorem 1.4] gives sufficient conditions for a self-similar tiling dynamical system tohave a pure discrete spectrum. This result implies that the diffraction measureγ of anyDelone setX constructed from a tiling in the dynamical system by marking tiles is a purediscrete measure. It seems likely that when these conditions are satisfied, the associatedDelone setX must be a Meyer set.

Geometric Models for Quasicrystals, I 189

Acknowledgments

I am indebted to Michael Baake, Ludwig Danzer, Richard Kenyon, Doug Lind, PeterPleasants, Jim Reeds, Marjorie Senechal, Boris Solomyak, Paul Wright, and the refer-ees for helpful comments. Martin Schlottman supplied a correction concerning Voronoitesselations on an earlier version of the paper, and suggested including Delaunay tesse-lations in Theorem 2.2.

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