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A Characterization of Model Multi-colour Sets

Model sets are always Meyer sets, but not vice-versa. This article is about characterizing model sets (general and regular) amongst the Meyer sets in terms of two associated dynamical systems. These two dynamical systems describe two very different topologies on point sets, one local and one global. In model sets these two are strongly interconnected and this connection is essentially definitive. The paper is set in the context of multi-colour sets, that is to say, point sets in which points come in a finite number of colours, that are loosely coupled together by finite local complexity.


Introduction
Model sets, or cut and project sets, are an essential ingredient in the study of long-range aperiodic order and in both the theoretical and experimental sides of quasicrystal theory. Model sets are Delone subsets of space and under mild assumptions display such nice properties as finite local complexity, repetitivity, and pure point diffractivity while at the same time being free of any translational symmetry. However, it is not easy to characterize these sets. By definition a model set, say in real space R n , is the projection of part of a lattice from some 'super-space' R n × H, where H is some locally compact Abelian group, the part to be projected being determined by some suitable compact set W in H. For a model set given to us in R n , all this additional baggage is hidden. The process of reconstructing it is part of the problem.
In the study of aperiodic order one needs to decide what it means for two aperiodic sets to look like each other. There are two very distinct ways in which this is commonly done. The first is to compare the two sets locally and see if, perhaps after a small translational shift, they agree on some large patch of space, say on a large ball around the origin. This leads to a topology on discrete sets which we call here the local topology. On the other hand one may take the view that it is not the local structure that is important, but rather the average structure. This leads us to determine that two discrete sets are close if, again perhaps after a small translational shift, the density of the symmetric difference of the two sets is small. This leads to the autocorrelation topology.
Starting with a discrete set Λ one forms the translational orbit R n + Λ. The closures of this orbit in the local and autocorrelation topologies lead to two dynamical hulls, X(Λ) and A(Λ). There is no reason to expect there to be any relationship between these two hulls since there is no inherent relationship between the local and autocorrelation topologies. But for regular model sets there is. In fact there is a continuous surjective R n -map β : X(Λ) −→ A(Λ), the so-called torus parameterization [13]. This remarkable confluence of the two topologies seems to be a fundamental property of quasi-crystal theory. In this paper we show that, in the context of Meyer sets (see below), it characterizes the model sets.
The setting of the paper is not just sets, but more generally multi-colour sets, or multi-sets as we will refer to them henceforth. 1 These are sets in which points are coloured by some finite number m of colours, so that in effect we have m different types of points. This generalization is rather natural for any attempts to model physical structures by using points to designate atomic positions, and it is likewise essential in the study of aperiodic tilings, in which we represent tiles of different types by suitably coloured points. It is particularly relevant for substitution tilings where the substitution process can only be formulated at the level of point sets by the use of multi-sets.
The second generalization we make is to replace R n by any compactly generated locally compact Abelian group G. This produces virtually no additional complications.
Briefly the main theorem of the paper can be stated in the form: Then there is a continuous G-map β : X(Λ) → A(Λ) which is one-to-one a.e. with respect to A(Λ) if and only if Λ (or equivalently each element of X(Λ)) is a regular model multi-set.
This paper derives much from a longer work [1]. That work is primarily concerned with characterizing model sets in terms of dynamical systems and their spectra, which is a considerably more complex story. Here our emphasis is to give a relatively short self-contained characterization of model sets in terms of the 'coincidence' of two very different topologies, and to do this not only for model sets, but also for model multi-sets.
The present paper also relies on the earlier work of Martin Schlottmann [13], and in fact the last section is essentially a reworking of his results, although the hypotheses are slightly different and again we are generalizing to multi-sets. In dealing with model sets, the devil is truly in the details. Model sets are in principle easy to understand, but the virtually unlimited scope of the defining window W produces all sorts of complications, and there are various definitions of model sets in the literature, depending on exactly which conditions are imposed on W . One of our contributions here is to find a slightly different (and basical more general) definition for model sets (Def. 2.5), which provides for a smoother characterization than is otherwise possible and allows an efficient and fairly self-contained proof of the characterization of model sets. The underlying context of the characterization is that of Meyer sets (Def. 2.1).

Preliminaries
Let G be a compactly generated locally compact Abelian group. A multi-set or m-multi-set in G is a subset Λ = Λ 1 × · · · × Λ m ⊂ G × · · · × G (m copies) where Λ i ⊂ G. We also write Λ = (Λ 1 , . . . , Λ m ) = (Λ i ) i≤m . Although Λ is a product of sets, it is convenient to think of it as a set whose points come in various 'types' or 'colours', i being the colour of the points in Λ i . A cluster of Λ is, by definition, a family P = (P i ) i≤m where P i ⊂ Λ i is finite for all i ≤ m. Many of the clusters that we consider have the form A ∩ Λ := (A ∩ Λ i ) i≤m , for a compact set A ⊂ G. There is a natural translation G-action on the set of multi-sets and their clusters in Λ. The translate of a cluster P by x ∈ G is x + P = (x + P i ) i≤m . We say that Λ is locally finite if for any compact set K in G, K ∩ Λ is finite (equivalently each Λ i is discrete and closed). A Delone set is a relatively dense and uniformly discrete subset in G and we say that Λ is Delone if each component Λ i is Delone. Throughout this paper, all multi-sets that we consider will be assumed to be locally finite.

Definition 2.3
A locally finite multi-set Λ is repetitive if for every compact set K ⊂ G, {t ∈ G : Λ ∩ K = (t + Λ) ∩ K} is relatively dense; i.e. there exists a compact set B K ∈ G such that every translate of B K contains at least one element of {t ∈ G : Λ ∩ K = (t + Λ) ∩ K}.
In order to speak about densities and autocorrelations, we need a way to specify averages. For this purpose we fix, once and for all, an averaging sequence A = {A n } n∈N satisfying (i) each A n is a compact set of G; (ii) for all n, A n ⊂ A • n+1 ; 2 All model sets are Meyer sets, but the reverse is far from true. Unlike model sets, Meyer sets have many characterizations [7]. Another definition, which is more commonly found in the literature, is the characterization of Jeffrey Lagarias: S is Meyer if it is Delone and S − S is uniformly discrete [5]. The proof in [5] is for real spaces. A proof for compactly generated locally compact Abelian groups can be found in [1]. Outside this setting the two concepts are different.
(iii) n∈N A n = G ; (iv) (the van Hove property) for all compact sets K ⊂ G, is the K-boundary of A n and θ is a Haar measure of G.
Definition 2.4 A cut and project scheme (CPS) consists of a collection of spaces and mappings as follows; where G is a compactly generated locally compact Abelian group, H is a locally compact Abelian group, π 1 and π 2 are the canonical projections, L ⊂ G × H is a lattice, i.e. a discrete subgroup for which the quotient group (G × H)/ L is compact, π 1 | L is injective, and π 2 ( L) is dense in H.
The model set Γ is regular if the boundary ∂W = W \W • of W is of (Haar) measure 0. We say that Γ is a model multi-set (resp. regular model multi-set) if each Γ i is a model set (resp. regular model set) with respect to the same CPS.
One should note here that as π 2 need not be 1 − 1 on L, the model set Γ need not actually be of the form Λ(V ) for any set V ⊂ H. Nonetheless it is hemmed in between two such sets differing only by points on the boundary of the window W . The reason for this slight relaxation in the definition of model sets is to allow for the assumption of irredundancy (5.3) later in the paper.
In talking about a model set there is always an implied CPS from which it arises. When we need to be more precise we explicitly mention the CPS.

Two dynamical hulls
In this section we introduce two dynamical hulls; one with a topology based on local structure and the other with a topology based on the average overall structure.
For the rest of the paper, Λ is a locally finite m-coloured multi-set for which each Λ i is relatively dense.
Define L to be the subgroup of G generated by all the sets Λ i : i ≤ m. This group is countable. In fact since G is compactly generated it is also σ-compact: G = ∞ n=1 K n where each K n is compact and K n ⊂ K n+1 for all n. Now Λ i is locally finite, so Λ i ∩ K n is finite for all n. Thus Λ i ∩ K n is countable, whence also Λ i and L.
Let D = D(m) be the set of all locally finite m-multi-sets in G. We set, for each compact set K ⊂ G and neighbourhood V of 0 in G, The set of U K,V forms a fundamental set of entourages of a uniformity on D for which the sets of the form We call this topology the local topology. The set D is complete with respect to the uniformity (see [13]) and is a Hausdorff space. The local hull X(Λ) is defined as the closure of the G-orbit of Λ in D, i.e. X(Λ) := {g + Λ : g ∈ G}.
Now we construct the autocorrelation group A(Λ). Let Λ ′ , Λ ′′ be two locally finite m-multi-sets in D. We define Here △ is the symmetric difference operator and {A n } is our averaging sequence.
For each open neighbourhood V of 0 in G and each ǫ > 0, define Then U forms a fundamental set of entourages for a uniformity on G. Since each U (V, ǫ) is G-invariant, we obtain a topological group structure on G. We will call it the autocorrelation topology. Let A(Λ) be the Hausdorff completion of G in this topology, which is a new topological group (see [3, Chapter III, The meaning of A(Λ) can be described as follows. The pseudo-metric d is defined on D. Define an equivalence relation on D by Let D := D/ ≡ and let d also denote the resulting G-invariant metric on D.
Then D is a complete space [9] and its elements can be identified with actual multi-sets in G up to density 0 changes. Now following the idea of the local topology we can put a new uniformity on D, which mixes the d-topology with the standard topology of G, by using the sets Then D is complete in this topology too. Just as in forming the local hull, we can start with Λ and form the closure A ′ of the orbit G + Λ in D. The mapping G → D, t → t + Λ allows us to pull the topology back to G. This is the content of (3.3). The A(Λ) is then G-homeomorphic with hull A ′ through the continuous extension of t → t + Λ. For this reason we may refer to A(Λ) as the autocorrelation hull. For The following result is a generalization of a result found in [9].
Proposition 3.1 Let Λ be a locally finite multi-set in G. Then A(Λ) is compact if and only if for all ǫ > 0, P ǫ is relatively dense in G.
Conversely, we assume that P ǫ is relatively dense for all ǫ > 0. Let ǫ > 0 and V ′ be an open neighbourhood of 0 in G. From the assumption, G ⊂ P ǫ + K ′ for some compact set K ′ . We can cover Hence G is precompact and A(Λ) is compact.
Let us assume that P ǫ is relatively dense for all ǫ > 0. Recall the subgroup L generated by all the points Λ i , i ≤ m. It is immediate that P ǫ ⊂ L for any ǫ < 2d(Λ, ∅). We can define a uniformity on L using the pseudo-metric d. Of course the {P ǫ } form a basis for a fundamental system of neighbourhoods of 0 in the corresponding topology. We define H to be a (Hausdorff) completion of L in this uniformity. Then H is a locally compact Abelian group (see [2]). By definition there exists a uniformly continuous mapping φ : L → H such that φ(L) is dense in H. Now we can construct a cut and project scheme: Here L is relatively dense and a discrete subgroup in G × H, and so the factor group T(Λ) := (G × H)/ L is compact (see [2]). The pseudo-metric d on L determines a corresponding metric d H on H. Let B H ǫ denote the corresponding open ball of radius ǫ in H. Then it is a standard feature of completions that φ(P ǫ ) = φ(L) ∩ B H ǫ [3]. We define Proposition 3.2 Let Λ be a locally finite multi-set in G. Suppose that P ǫ is relatively dense for all ǫ > 0. Then A(Λ) ∼ = T(Λ).
Proof. T(Λ) may be viewed as the completion of G under the uniform topology that is the coarsest topology on G for which the map ι : This is the same topology on G as the autocorrelation topology on G. Thus A(Λ) ∼ = T(Λ).

Corollary 3.3
Let Λ be a multi-set with FLC. Suppose that there exists a continuous G-map β : Thus P ǫ is relatively dense in G for all ǫ > 0. By Proposition 3.2 we can conclude that A(Λ) ∼ = T(Λ).
Thus we may identify A(Λ) with T(Λ) when it is convenient. Proof. Choose any ǫ > 0. Since each Λ i is a Meyer set, Λ is locally finite and there exist finite sets From the assumption, for any given P ǫ we can find a compact set K ǫ such that G ⊂ P ǫ + K ǫ . So ∆ i ⊂ P ǫ + K ǫ for each i ≤ m. Since we are assuming that each Λ i is relatively dense, for small enough ǫ > 0, P ǫ ⊂ ∩ m i=1 ∆ i . Then by the assumption where N i is a finite subset of L. Thus ∆ i ⊂ P ǫ + N i for each i ≤ m. So ∆ i is precompact (totally bounded) in L with the pseudo-metric topology and φ(∆ i ) is compact in H for each i ≤ m.
{Γ}. The set of non-singular elements of X(Λ) consists of G-orbits. In this section we show that the existence of a torus parmetrization together with the existence of a non-singular element in X(Λ) ensures that Λ is a model multi-set.
Proposition 4.1 Let Λ be a multi-set in G with finite local complexity (FLC). Suppose that there exists a continuous G-map β : X(Λ) → A(Λ). If Γ is nonsingular in X(Λ), then given any Proof. Suppose that there is a positive integer M such that for every n > 0 there exists t n ∈ P (1/2 n ) for which Since X(Γ) ⊂ X(Λ) is compact from the FLC of Λ, {t n + Γ} n has a convergent subsequence {t n k + Γ} k such that t n k + Γ → Γ ′ . On the other hand, identifying A(Λ) with T(Λ), Then β(Γ) = β(Γ ′ ), and by the assumption that Γ is non-singular in X(Λ) Γ = Γ ′ . It follows that t n k + Γ  Proof. Since Γ ∈ X(Λ), we can choose s ∈ G and a compact set K for which (s + Γ) ∩ K = Λ ∩ K and each (s + Γ i ) ∩ K = ∅. Then s + Γ i ⊂ Λ i + L ⊂ L for all i ≤ m. Furthermore, since β is G-map and Γ is non-singular, s + Γ is also non-singular. So we can translate and assume at the outset that supp(Γ) ⊂ L.
Remark: By this last Proposition it is possible to make the sets U i a little nicer, replacing them by W i • where W i := U i , since these differ only from the U i by boundary points, and these additional points do not affect the model multi-set Γ. This W i has the nice property that W i is the closure of its interior, a property often assumed in defining model sets.
The following proposition holds in general CPS.
with respect to some CPS (see (2.1)). Then Λ has FLC, and for any Γ ∈ X(Λ) there exists (−s, −h) ∈ G × H so that Proof. For each Γ ∈ X(Λ) we can choose s ∈ G such that s Note that W := i≤m V i is compact. So for any compact set K ⊂ G, (Λ(W ) − Λ(W )) ∩ K has a finite number of elements. This implies that for any x ∈ supp(Λ) there are, up to translation, only finitely many different subsets of the form (K + x) ∩ Λ. Therefore the claim follows. Then X(Λ) is compact and we can find {t n } n ⊂ L such that {t n + Λ} converges to s + Γ in X(Λ). There exists n 0 ∈ Z + such that for any k, l ≥ n 0 , (t k + Λ) ∩ (t l + Λ) = ∅ and so is compact, we can find a convergent subsequence of {φ(t n )} n . Without loss of generality, we can assume that lim n→∞ φ(t n ) =: h ∈ H.
On the other hand, if z ∈ s+Γ i , then z ∈ t n +Λ i for large n. So φ(z) ∈ φ(t n )+V i for large n and φ(z) ∈ h + V i . Thus z ∈ Λ(h + V i ). Therefore Note that β(t n + Λ) = ι(t n ) + β(Λ) and ι(t n ) = (t n , 0) + L = (0, −φ(t n )) + L. Thus Corollary 4.5 Let Λ be a multi-set with FLC. Suppose that there exists a continuous G-map β : X(Λ) → A(Λ), and Λ i = Λ(W i ) with respect to CPS (3.4) where W i ⊂ H is compact and Proof. If Γ is non-singular in X(Λ), from Proposition 4.2 and Proposition Proposition 4.6 Let Λ be a multi-set in G with FLC and repetitivity. Suppose that there exists a continuous G-map β : and ∂V i has empty interior for each i ≤ m, with respect to CPS (3.4). Then there exists a non-singular element Λ ′ in X(Λ) such that Λ ′ i = Λ(W i ) where W i is compact and W i = W i • for each i ≤ m with respect to the same CPS (3.4), and so that for each Γ ∈ X(Λ) there exists Proof. From Proposition 4.4, for any Γ ∈ X(Λ) there exists (−s, −h) ∈ G × H so that By the Baire category theorem, there exists h ′ ∈ H such that Since repetitivity of Λ is equivalent to the minimality of X(Λ) [4,13], X(Λ) = X(Λ ′ ). Also Λ ′ is non-singular by construction. We conclude using Proposition 4.4 again.
Proposition 4.7 Let Λ be a multi-set with FLC. Suppose that there exists a continuous G-map β : X(Λ) → A(Λ) which is one-to-one a.e. A(Λ), and where θ H is any Haar measure in H. We note that if Λ is a Meyer multi-set then Λ has FLC.

Proof. From
In other words, for each Γ ∈ X(Λ), Γ is a regular model multi-set.

Torus parameterizations from model multisets
We consider a cut and project scheme : where H is a locally compact Abelian group, π 1 and π 2 are canonical maps, L is a lattice in G × H, π 1 | L is one-to-one, and π 2 ( L) is dense in H. Let L = π 1 ( L). We define φ : L → H by φ(x) = π 2 (π −1 1 (x)). Suppose that Λ is a multi-set in G and that for each i ≤ m, There are two things that we can do to tighten up the cut and project scheme without altering the multiset Λ.
First we can require that the group H 0 := W i : i ≤ m is all of H. To see this, note first that the subgroup of H generated by the W i • is an open, hence closed subgroup of H, and so it contains all the W i and must then be H 0 . So H 0 is closed. Let L 0 := L ∩ (G × H 0 ) and L 0 its π 1 projection. Now we can check that we can replace H, L by H 0 , L 0 in the cut and project scheme to get another one with the desired property.
Once we know that H is generated by the windows, then it follows that Then for any coset x + Λ i : i ≤ m ⊂ L, φ(x) + φ( Λ i : i ≤ m ) is also dense in H and so has a point φ(x) + φ(u) ∈ W • 1 , say, where u ∈ Λ i : i ≤ m . Since x + u ∈ L, it is in fact in Λ 1 and so we obtain x ∈ Λ i : i ≤ m .
A consequence of this is that, as we saw at the start of Sec. 3, L, and so also L, is now countable.
Let I := {t ∈ H : t + W i = W i for all i ≤ m}. Translations t in H of this form indicate certain redundancy in H. The second way in which we can tighten up the CPS is to remove this redundancy in H by factoring out the subgroup I. We define H ′ := H/I, ψ : L → H ′ by ψ(x) = φ(x) + I, and Let W ′ i denote W i + I in H ′ . Then we can construct a new cut and project scheme : [11,Theorem 3.3.28]).
Thus without loss of generality we will assume both that H is generated by the windows of Λ and a situation that we will refer to as irredundancy. All subsequent CPS will be assumed reduced into this form.
In this section we establish the existence of the 'torus parametrization'. The following theorem is proved after a sequence of auxiliary propositions.
and θ H (∂W i ) = 0 for all i ≤ m with respect to some irredundant CPS. Then there is a continuous G-map β : X(Λ) → A(Λ) which is one-to-one a.e. A(Λ).
The rest follows directly from Proposition 5.2.
Each of (−φ(t) + W i )\U ′ is closed and so there exists a compact K in G with (s + Γ) ∩ K = ∅ such that This shows Therefore γ is continuous. Furthermore, The proofs of the existence of the torus parametrization shown here are essentially due to Schlottmann [13]. There he proves the existence of the map under the condition that a point set Λ satisfies Λ(W • ) ⊂ Λ ⊂ Λ(W ), where W is compact, and is repetitive (instead of assuming that the window W is the closure of its interior). In fact it is proved in [13] that the repetitivity of the point set implies that the window is the closure of its interior. However they are not equivalent conditions. Here we assume that each of the windows is the closure of its interior and we also place his results into multi-set setting. We just point out here that in the multi-set setting the existence of the torus parametrization can also be proved under the assumption of repetitivity of a multi-set satisfying Λ(W i where W i is compact, without the assumption that each of the windows is the closure of its interior. Let X g := {Γ ∈ X(Λ) : φ(L) ∩ (c Γ + m i=1 ∂W i ) = ∅}. Note that X g = ∅ by the Baire category theorem.
Proposition 5.5 Let Λ be a multi-set in G. Suppose that Λ(W i • ) ⊂ Λ i ⊂ Λ(W i ), W i is compact, and W i = W i • = ∅ for all i ≤ m with respect to some irredundant CPS. Then γ| Xg is one-to-one.
Proposition 5.6 Let Λ be a multi-set in G. Suppose that Λ(W i • ) ⊂ Λ i ⊂ Λ(W i ), W i is compact, and W i = W i • = ∅ for all i ≤ m, with respect to some irredundant CPS. Then T g = T(Λ)\((G × m i=1 ∂W i + L)/ L).
Proof. Note that ∂W i + L)/ L. Proof. We will show that λ((G × m i=1 ∂W i + L)/ L) = 0. By [11,Theorem 3.3.28], we only need to show that ν(G × m i=1 ∂W i + L) = 0, where ν is a Haar measure in G × H. Since G is compactly generated, there exists a sequence of compact sets {K n } such that G = ∞ i=1 K n . Each ν(K n × m i=1 ∂W i ) = 0 from the assumption that θ H (∂W i ) = 0 for all i ≤ m. Since L is countable, ν(G × m i=1 ∂W i + L) = 0. Thus the assertion follows. The following proposition is a modification of [9, Prop. 5.1] for multi-sets. Proof. A(Λ) is the completion of G under the autocorrelation topology. T(Λ) may be considered as the completion of G when it is given the coarsest topology for which the mapping x → (x, 0)+ L of G into T(Λ) is continuous. It will suffice to show that these two topologies on G are the same. If x ∈ G is close to 0 in T-topology, then for small open neighbourhoods V of 0 in G and V 1 of 0 in H there exists (t, φ(t)) ∈ L such that x−t ∈ V and φ(t) ∈ V 1 . On the other hand if x ∈ G is close to 0 in A-topology, then for a small open neighbourhood U of 0 in G and some ǫ > 0 there exists t ∈ L such that x − t ∈ U and d(t + Λ, Λ) < ǫ. So we need to show that for t ∈ L, φ(t) is close to 0 in H if and only if d(t + Λ, Λ) is close to 0.
For t ∈ L,