Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings

The equivalence between pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings is discussed.

Abstract

Primitive substitution tilings on whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

1. Introduction  

In the study of aperiodic tilings, it has been an interesting problem to characterize pure discrete spectrum of tiling dynamical systems (Baake & Moody, 2004 ▸). This property is related to understanding the structure of mathematical quasicrystals. For this direction of study, substitution tilings have been good models, since they have highly symmetrical structures. A lot of research has been done in this direction (see Akiyama et al., 2015 ▸; Baake & Grimm, 2013 ▸ and references therein). Given a substitution tiling with pure discrete spectrum, it is known that this can be described via a cut-and-project scheme (CPS) (Lee, 2007 ▸). However, in the work of Lee (2007 ▸), the construction of the CPS is with an abstract internal space built from the pure discrete spectral property. Since the internal space is an abstract space, it is neither easy to understand the tiling structure, nor clear if the model sets are regular or not. In the case of one-dimensional substitution tilings with pure discrete spectrum, it is shown that a CPS with a Euclidean internal space can be built and the corresponding representative point sets are regular model sets (Barge & Kwapisz, 2006 ▸). In this paper, we consider substitution tilings on with pure discrete spectrum whose expansion maps are unimodular. We show that it is possible to construct a CPS with a Euclidean internal space and that the corresponding representative point sets are regular model sets in that CPS.

The outline of the paper is as follows. First, we consider a repetitive primitive substitution tiling on whose expansion map is unimodular. Then we build a CPS with a Euclidean internal space in Section 3. In Section 4, we discuss some known results around pure discrete spectrum, Meyer set and Pisot family. In Section 5, under the assumption of pure discrete spectrum, we show that each representative point set of the tiling is actually a regular model set in the CPS with a Euclidean internal space.

2. Preliminaries  

2.1. Tilings  

We begin with a set of types (or colours) , which we fix once and for all. A tile in is defined as a pair where (the support of T) is a compact set in , which is the closure of its interior, and is the type of T.

We let for . We say that a set P of tiles is a patch if the number of tiles in P is finite and the tiles of P have mutually disjoint interiors. The support of a patch is the union of the supports of the tiles that are in it. The translate of a patch P by is . We say that two patches and are translationally equivalent if for some . A tiling of is a set of tiles such that and distinct tiles have disjoint interiors. We always assume that any two -tiles with the same colour are translationally equivalent (hence there are finitely many -tiles up to translations). Given a tiling , a finite set of tiles of is called a -patch. Recall that a tiling is said to be repetitive if every -patch occurs relatively densely in space, up to translation. We say that a tiling has finite local complexity (FLC) if, for every R > 0, there are finitely many equivalence classes of -patches of diameter less than R.

2.2. Delone κ-sets  

A κ-set in is a subset (κ copies) where and κ is the number of colours. We also write . Recall that a Delone set is a relatively dense and uniformly discrete subset of . We say that is a Delone κ-set in if each is Delone and is Delone.

The types (or colours) of points for Delone κ-sets have a meaning analogous to the colours of tiles for tilings. We define repetitivity and FLC for a Delone κ-set in the same way as for tilings. A Delone set Λ is called a Meyer set in if is uniformly discrete, which is equivalent to saying that for some finite set F (see Moody, 1997 ▸). If is a Delone κ-set and ) is a Meyer set, we say that is a Meyer set.

2.3. Substitutions  

We say that a linear map is expansive if there is a constant c > 1 with

for all under some metric d on compatible with the standard topology.

Definition 2.1   —

Let be a finite set of tiles on such that ; we will call them prototiles. Denote by the set of patches made of tiles each of which is a translate of one of ’s. We say that is a tile-substitution (or simply substitution) with an expansive map ϕ if there exist finite sets for , such that

with

Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the to be empty. We call the finite set a digit set (Lagarias & Wang, 1996 ▸). The substitution matrix of the tile-substitution is defined by . We say that ϕ is unimodular if the minimal polynomial of ϕ over has constant term (i.e. ); that is to say, the product of all roots of the minimal polynomial of ϕ is .

Note that for

where

The tile-substitution is extended to translated prototiles by

The equations (2) allow us to extend ω to patches in defined by . It is similarly extended to tilings all of whose tiles are translates of the prototiles from . A tiling satisfying is called a fixed point of the tile-substitution, or a substitution tiling with expansion map ϕ. It is known that one can always find a periodic point for ω in the tiling dynamical hull, i.e. for some . In this case we use in the place of ω to obtain a fixed-point tiling. We say that the substitution tiling is primitive, if there is an for which has no zero entries, where is the substitution matrix.

Definition 2.2   —

is called a substitution Delone κ-set if is a Delone κ-set and there exist an expansive map and finite sets for such that

where the unions on the right-hand side are disjoint.

There is a standard way to choose distinguished points in the tiles of a primitive substitution tiling so that they form a ϕ-invariant Delone κ-set. They are called control points (Thurston, 1989 ▸; Praggastis, 1999 ▸) which are defined below.

Definition 2.3   —

Let be a fixed point of a primitive substitution with an expansion map ϕ. For every -tile T, we choose a tile on the patch ; for all tiles of the same type, we choose with the same relative position. This defines a map called the tile map. Then we define the control point for a tile by

The control points satisfy the following:

(a) , for any tiles of the same type;

(b) , for .

For tiles of any tiling , the control points have the same relative position as in -tiles. The choice of control points is non-unique, but there are only finitely many possibilities, determined by the choice of the tile map. Let

It is possible to consider a tile map

Then for any ,

Let

be a set of control points of the tiling in . In what follows, if there is no confusion, we will use the same notation to mean .

For the main results of this paper, we need the property that with . Under the assumption that ϕ is unimodular, this can be achieved by taking a proper control point set which comes from a certain tile map. We define the tile map as follows. It is known that there exists a finite patch in a primitive substitution tiling which generates the whole tiling (Lagarias & Wang, 2003 ▸). Although it was defined with primitive substitution point sets by Lagarias & Wang (2003 ▸), it is easy to see that the same property holds for primitive substitution tilings. We call the finite patch the generating tile set. When we apply the substitution infinitely many times to the generating tile set , we obtain the whole substitution tiling. So there exists such that nth iteration of the substitution to the generating tile set covers the origin. We choose a tile R in a patch which contains the origin, where for some . Then there exists a fixed tile such that . Replacing the substitution ω by , we can define a tile map γ so that

Then by the definition of the control point sets and so . Notice that

since ϕ is unimodular. From the construction of the tile map, we have for any . From (9), for any . Hence . Thus

Remark 2.4   —

In the case of primitive unimodular irreducible one-dimensional Pisot substitution tilings, it is known that by choosing the left end points of the tiles as the control points (see Barge & Kwapisz, 2006 ▸; Sing, 2007 ▸).

2.4. Pure point spectrum and algebraic coincidence  

Let be the collection of tilings on each of whose patches is a translate of a -patch. In the case that has FLC, we give a usual metric δ on the tilings in such a way that two tilings are close if there is a large agreement on a big region with small shift (see Schlottmann, 2000 ▸; Radin & Wolff, 1992 ▸; Lee et al., 2003 ▸). Then

where the closure is taken in the topology induced by the metric δ. For non-FLC tilings, we can consider ‘local rubber topology’ on the collection of tilings (Müller & Richard, 2013 ▸; Lenz & Stollmann, 2003 ▸; Baake & Lenz, 2004 ▸; Lee & Solomyak, 2019 ▸) and obtain as the completion of the orbit closure of under this topology. For tilings with FLC, the two topologies coincide. In the case of either FLC or non-FLC tilings, we obtain a compact space . We have a natural action of on the dynamical hull of by translations and get a topological dynamical system . Let us assume that there is a unique ergodic measure μ in the dynamical system and consider the measure-preserving dynamical system . It is known that a dynamical system with a primitive substitution tiling always has a unique ergodic measure (Solomyak, 1997 ▸; Lee et al., 2003 ▸). We consider the associated group of unitary operators on :

Every defines a function on by . This function is positive definite on , so its Fourier transform is a positive measure on called the spectral measure corresponding to g. The dynamical system is said to have pure discrete spectrum if is pure point for every . We also say that has pure discrete spectrum if the dynamical system has pure discrete spectrum.

When we restrict discussion to primitive substitution tilings, we note that a tiling has pure discrete spectrum if and only if the control point set of the tiling admits an algebraic coincidence (see Definition 2.5). So from now on when we assume pure discrete spectrum for , we can directly use the property of algebraic coincidence. We give the corresponding definition and theorem below.

Definition 2.5   —

Let be a primitive substitution tiling on with an expansive map ϕ and be a corresponding control point set. We say that admits an algebraic coincidence if there exists and for some such that

Here recall from (7) that .

Note that, if the algebraic coincidence is assumed, then for some

Theorem 2.6   —

[Theorem 3.13 (Lee, 2007 ▸)] Let be a primitive substitution tiling on with an expansive map ϕ and be a corresponding control point set. Suppose that all the eigenvalues of ϕ are algebraic integers. Then has pure discrete spectrum if and only if admits an algebraic coincidence.

The above theorem is stated with FLC by Lee (2007 ▸). But from Lemma 4.1 and Proposition 4.2, pure discrete dynamical spectrum of implies the Meyer property of the control point set . All Meyer sets have FLC. So it is a consequence of pure discrete dynamical spectrum. On the other hand, the algebraic coincidence implies that

This means that is uniformly discrete and thus Ξ is uniformly discrete. From Moody (1997 ▸), we obtain that is uniformly discrete. For any ,

Hence is a Meyer set (Moody, 1997 ▸). Thus it is not necessary to assume FLC here. There is a computable algorithm to check the algebraic coincidence in a primitive substitution tiling (Akiyama & Lee, 2011 ▸).

2.5. Cut-and-project scheme  

We give definitions for a CPS and model sets constructed with and a locally compact Abelian group.

Definition 2.7   —

A cut-and-project scheme (CPS) consists of a collection of spaces and mappings as follows:

where is a real Euclidean space, H is a locally compact Abelian group, and are the canonical projections, is a lattice, i.e. a discrete subgroup for which the quotient group is compact, is injective and is dense in H. For a subset , we denote

A model set in is a subset of of the form , where has non-empty interior and compact closure. The model set is regular if the boundary of W

is of (Haar) measure 0. We say that is a model κ-set (respectively, regular model κ-set) if each is a model set (respectively, regular model set) with respect to the same CPS. Especially when H is a Euclidean space, we call the model set Λ a Euclidean model set (see Baake & Grimm, 2013 ▸).

3. Cut-and-project scheme on substitution tilings  

Throughout the rest of the paper, we assume that ϕ is diagonalizable, the eigenvalues of ϕ are algebraically conjugate with the same multiplicity, since the structure of a module generated by the control points is known only under this assumption (Lee & Solomyak, 2012 ▸).

Let

be the distinct real eigenvalues of ϕ and

be the distinct complex eigenvalues of ϕ. By the above assumption, all these eigenvalues appear with the same multiplicity, which we will denote by J. Recall that ϕ is assumed to be diagonalizable over . For a complex eigenvalue λ of ϕ, the diagonal block

is similar to a real matrix

where , and

Since ϕ is diagonalizable, by eventually changing the basis in , we can assume without loss of generality that

where is a real matrix for , a real matrix of the form

for , and is the zero matrix, and .

Let . Note that m is the degree of the minimal polynomial of ϕ over . For each , let

Further, for each we have the direct sum decomposition

such that each is -invariant and , identifying with or .

Let .

Let be the canonical projection of onto such that

where and with .

We define such that for each ,

We recall the following theorem for the module structure of the control point sets. From Lemma 6.1 (Lee & Solomyak, 2012 ▸), we can readily obtain the property:1

which is used in the proof of Lemma 5.2. So we state Theorem 4.1 (Lee & Solomyak, 2012 ▸) in the following form. Let

Theorem 3.1   —

[Theorem. 4.1 (Lee & Solomyak, 2012 ▸)] Let be a repetitive primitive substitution tiling on with an expansion map ϕ. Assume that has FLC, ϕ is diagonalizable, and all the eigenvalues of ϕ are algebraically conjugate with the same multiplicity J. Then there exists an isomorphism such that

where , , are given in (18), and .

Since ϕ is a block diagonal matrix as shown in (16), we can note that are linearly independent over .

A tiling is said to be rigid if satisfies the result of Theorem 3.1; that is to say, there exists a linear isomorphism such that

where , , are given in (18). One can find an example of a non-FLC tiling that the rigidity property fails in (Frank & Robinson, 2008 ▸; Lee & Solomyak, 2019 ▸).

3.1. Construction of a cut-and-project scheme  

Consider that ϕ is unimodular and diagonalizable, all the eigenvalues of ϕ are algebraic integers and algebraically conjugate with the same multiplicity J, and is rigid. Since ϕ is an expansion map and unimodular, there exists at least one other algebraic conjugate other than eigenvalues of ϕ. Under this condition, we construct a CPS with a Euclidean internal space. In the case of multiplicity 1, the CPS was first introduced in Lee et al. (2018 ▸). For earlier development, see Siegel & Thuswaldner (2009 ▸).

It is known that if ϕ is a diagonalizable expansion map of a primitive substitution tiling with FLC, every eigenvalue of ϕ is an algebraic integer (Kenyon & Solomyak, 2010 ▸). So it is natural to assume that all the eigenvalues of ϕ are algebraic integers in the assumption. In (16), suppose that the minimal polynomial of ψ over has e number of real roots and f number of pairs of complex conjugate roots. Recall that

are distinct eigenvalues of ϕ from (13) and (21). Let us consider the roots in the following order:

for which

where are the same as in (13) and (14).

Let

We consider a space where the rest of the roots of the minimal polynomial of ψ other than the eigenvalues of ψ lie. Using similar matrices as in (15) we can consider the space as a Euclidean space. Let

For , define a matrix

where is a real matrix with the value for , and is a real matrix of the form

for . Notice that ϕ and ψ have the same minimal polynomial over , since ϕ is the diagonal matrix containing J copies of ψ. Let us consider now the following algebraic embeddings:

where is a polynomial over and . Note that

Now we can define a map

Since are linearly independent over , the map Ψ is well defined. Thus for

where . Let .

Let us construct a CPS:

where and are canonical projections,

and

It is easy to see that is injective. We shall show that is dense in and is a lattice in . We note that is injective, since Ψ is injective. Since ϕ commutes with the isomorphism σ in Theorem 3.1, we may identify and its isomorphic image. Thus, from Theorem 3.1,

where . Note that for any and , . So we can note that

Lemma 3.2   —

is a lattice in .

Proof   —

By the Cayley–Hamilton theorem, there exists a monic polynomial of degree n such that . Thus every element of is expressed as a polynomial of ϕ of degree with integer coefficients where the constant term is identified with a constant multiple of the identity matrix. Therefore L is a free -module of rank nJ. Notice that L and are isomorphic -modules so that is also a free -module of rank nJ on . Let us define

Then, in fact, for any ,

Define also

which is a linear map on . Note that and is isomorphic to the image of by multiplication of the matrix . Since A is non-degenerate by the Vandermonde determinant, forms a basis of over . Thus is a lattice in .

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Lemma 3.3   —

and is dense in .

Proof   —

For simplicity, we prove the totally real case, i.e. for all i. Since the diagonal blocks of ϕ are all the same, it is enough to show that is dense in . By Theorem 24 (Siegel, 1989 ▸), is dense in if

implies for . The condition is equivalent to with in the terminology of Lemma 3.2. Multiplying by the inverse of A, we see that the entries of ξ must be Galois conjugates. As ξ has at least one zero entry, we obtain which shows for . In fact, this discussion is using the Pontryagin duality that the has a dense image if and only if its dual map is injective [see also Meyer (1972 ▸, ch. II, Section 1), Iizuka et al. (2009 ▸), Akiyama (1999 ▸)]. The case with complex conjugates is similar.

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Now that we have constructed the CPS (23), we would like to introduce a special projected set which will appear in the proofs of the main results in Section 5. For , we define

In the following lemma, we find an adequate window for a set and note that is a Meyer set.

Lemma 3.4   —

For any and , if , then

and forms a Meyer set.

Proof   —

Note that

Notice that if ϕ is unimodular, then and . Thus

It is easy to see that the set in (28) is contained in the set in (29). The inclusion for the other direction is due to the fact that and . Hence for any ,

Since (23) is a CPS and is bounded, forms a Meyer set for each (see Moody, 1997 ▸).

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4. Pure discrete spectrum, Meyer set and Pisot family  

Lemma 4.1   —

[Lemma 4.10 (Lee & Solomyak, 2008 ▸)] Let be a tiling on . Suppose that has pure discrete dynamical spectrum. Then the eigenvalues for the dynamical system span .

Proposition 4.2   —

[Proposition 6.6 (Lee & Solomyak, 2019 ▸)] Let be a primitive substitution tiling on with an expansion map ϕ. Suppose that all the eigenvalues of ϕ are algebraic integers. Assume that the set of eigenvalues of is relatively dense. Then is a Meyer set.

We note that ‘repetitivity’ is not necessary for Proposition 4.2. Under the assumption that is a primitive substitution tiling on , the following implication holds:

Definition 4.3   —

A set of algebraic integers is a Pisot family if for any , every Galois conjugate γ of , with , is contained in Θ. For , with real and , this reduces to being a real Pisot number, and for , with non-real and , to being a complex Pisot number.

Under the assumption of rigidity of , we can derive the following proposition from Lemma 5.1 (Lee & Solomyak, 2012 ▸) without additionally assuming repetitivity and FLC.

Proposition 4.4   —

[Lemma 5.1 (Lee & Solomyak, 2012 ▸)] Let be a primitive substitution tiling on with a diagonalizable expansion map ϕ. Suppose that all the eigenvalues of ϕ are algebraic conjugates with the same multiplicity and is rigid. Then if the set of eigenvalues of is relatively dense, then the set of eigenvalues of ϕ forms a Pisot family.

5. Main result  

We consider a primitive substitution tiling on with a diagonalizable expansion map ϕ. Suppose that all the eigenvalues of ϕ are algebraically conjugate with the same multiplicity J and is rigid. Additionally we assume that there exists at least one algebraic conjugate λ of eigenvalues of ϕ for which . Recall that

where is the set of control points of tiles of type i and . By the choice of the control point set in (10), we note that .

Lemma 5.1   —

Assume that the set of eigenvalues of ϕ is a Pisot family. Then for some , where is given in (26).

Proof   —

Since we are interested in Ξ which is a collection of translation vectors, the choice of control point set does not really matter. So we use the tile map (8) which sends a tile to the same type of tiles in . From Lemma 4.5 (Lee & Solomyak, 2008 ▸), for any ,

Since ϕ is an expansive map and satisfies the Pisot family condition, the maps and Ψ are defined with all the algebraic conjugates of eigenvalues of ϕ whose absolute values are less than 1. Thus for some . From the definition of in (26), .

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Lemma 5.2   —

Assume that has pure discrete spectrum. Then for any , there exists such that .

Proof   —

Note from (24) that for any and , is contained in Ξ. Recall that , where , . So any element is a linear combination of over . Applying (11) many times if necessary, we get that for any , for some .

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Proposition 5.3   —

Let be a primitive substitution tiling on with an expansion map ϕ. Under the assumption of the existence of CPS (23), if has pure discrete spectrum, then there exists such that

Proof   —

We first prove that there exists a finite set F such that for all , for some . This can be obtained directly from Lemma 5.5.1 (Strungaru, 2017 ▸; Baake & Grimm, 2017 ▸), but for the reader’s convenience we give the proof here. Note that is a Meyer set and for some . Since Ξ is relatively dense, for any , there exists such that . From the Meyer property of , the point set configurations

are finite up to translation elements of . We should note that if has FLC but not the Meyer property, the property (32) may not hold. Let

Then

, and F is a finite set. Thus for any ,

From Lemma 5.2 and , for any , there exists such that . By the pure discrete spectrum of and (11), there exists such that

Applying the containment (34) finitely many times, we obtain that there exists such that . Hence together with (33), there exists such that

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In order to discuss model sets and compute the boundary measures of their windows for substitution tilings, we need to introduce -set substitutions for substitution Delone sets which represent the substitution tilings.

Definition 5.4   —

For a substitution Delone κ-set satisfying (2), define a matrix whose entries are finite (possibly empty) families of linear affine transformations on given by . We define for . For a κ-set let

Thus by definition. We say that Φ is a κ-set substitution. Let

be a substitution matrix corresponding to Φ. This is analogous to the substitution matrix for a tile-substitution.

Recall that there exists a finite generating set such that

from Lagarias & Wang (2003 ▸), Lee et al. (2003 ▸). If the finite generating set consists of a single element, we say that is generated from one point. Since is dense in by Lemma 3.3, we have a unique extension of Φ to a κ-set substitution on in the obvious way; if for which , , we define , , D is given in (22), and . Since is dense in , we can extend the mapping to . If there is no confusion, we will use the same notation for the extended map.

Note that, by the Pisot family condition on ϕ, there exists some such that for any . This formula defines a mapping on and is a contraction on . Thus a κ-set substitution Φ determines a multi-component iterated function system on . Let be a substitution matrix corresponding to . Defining the compact subsets

and using (36) and the continuity of the mappings, we have

This shows that are the unique attractor of .

Remark 5.5   —

From Proposition 4.4 (Lee, 2007 ▸), if has pure discrete spectrum, then there exists such that the control point set of the tiling satisfies

for some , and . Note that . Let . We can consider a rth-level supertiling of . Note that there exists an rth-level supertile in containing the origin in the support which contains the tile . Redefining the tile map for the control points of this supertiling so that the control point of the rth-level supertile is at the origin, we can build a substitution tiling for which algebraic coincidence occurs at the origin. So rewriting the substitution if necessary, we can assume that . With this assumption, we get the following proposition.

Proposition 5.6   —

Let be a primitive substitution tiling on with a diagonalizable expansion map ϕ which is uni­modular. Suppose that all the eigenvalues of ϕ are algebraic conjugates with the same multiplicity and is rigid. Suppose that

for some , and . Assume that CPS (23) exists. Then each point set

is a Euclidean model set in CPS (23) with a window in which is open and pre-compact.

Proof   —

For each and , there exists such that

From ,

By Theorem 2.6 and Proposition 5.3, there exists such that . Thus

where depends on z. From the equality of (30), we let

Then

for any .

From Lemma 5.1, for some . Thus . Since is compact, is compact. Thus is compact.

□

We can assume that the open window in (39) is the maximal element satisfying (39) for the purpose of proving the following proposition. In this proposition, we show that the control point set is a regular model set using Keesling’s argument (Keesling, 1999 ▸).

Proposition 5.7   —

Let be a repetitive primitive substitution tiling on with a diagonalizable expansion map ϕ which is unimodular. Suppose that all the eigenvalues of ϕ are algebraic conjugates with the same multiplicity and is rigid. Under the assumption of the existence of CPS (23), if

where , and , then each Euclidean model set , has a window with boundary measure zero in the Euclidean internal space of CPS (23).

Proof   —

Let us define , where is the maximal open set in satisfying (39). From the assumption of (40), we first note that ϕ fulfils the Pisot family condition from Theorem 2.6 and Proposition 4.4. For every measurable set and for any with ,

where μ is a Haar measure in and D is the contraction as given in (22). Note that . In particular,

We have attractors ’s satisfying

Let us denote for and = . Then for any ,

Note here that for any , follows from the fact that has a non-empty interior. Thus

Note from Lagarias & Wang (2003 ▸) that the Perron eigenvalue of is . From the unimodular condition of ϕ,

Since is primitive, from Lemma 1 (Lee & Moody, 2001 ▸)

By the positivity of and , = .

Recall that for any ,

From (3), for any ,

and

Note that and is a non-empty open set. As , is dense in . Since is a Euclidean space, we can find a non-empty open set such that . So there exist and such that . Since ,

Thus there exists such that

Hence

The inclusion (43) is followed by the maximal choice of an open set . Let

Then

From (42), we observe that not all functions in are used for the inclusion (44). Thus there exists a matrix for which

where and . If , again from Lemma 1 (Lee & Moody, 2001 ▸), . This is a contradiction to (42). Therefore for any .

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The regularity property of model sets can be shared for all the elements in . One can find the earliest result of this property in the work of Schlottmann (2000 ▸) and the further development in the work of Baake et al. (2007 ▸), Keller & Richard (2019 ▸) and Lee & Moody (2006 ▸). We state the property [Proposition 4.4 (Lee & Moody, 2006 ▸)] here.

Proposition 5.8   —

(Schlottmann, 2000 ▸; Baake et al., 2007 ▸; Keller & Richard, 2019 ▸; Lee & Moody, 2006 ▸) Let be a Delone κ-set in for which where is compact and for with respect to some CPS. Then for any , there exists so that

From the assumption of pure discrete spectrum and Remark 5.5, we can observe that the condition (40) is fulfilled in the following theorem.

Theorem 5.9   —

Let be a repetitive primitive substitution tiling on with a diagonalizable expansion map ϕ which is unimodular. Suppose that all the eigenvalues of ϕ are algebraically conjugate with the same multiplicity. If has pure discrete spectrum, then each control point set , , is a regular Euclidean model set in CPS (23).

Proof   —

Under the assumption of pure discrete spectrum, we know that has FLC from the work of Lee & Solomyak (2019 ▸) and ϕ fulfils the Pisot family condition (Lee & Solomyak, 2012 ▸). From Theorem 3.1, we know that is rigid. Since ϕ is unimodular, there exists at least one algebraic conjugate λ of eigenvalues of ϕ for which . Thus we can construct the CPS (23) with a Euclidean internal space. Since has pure discrete spectrum and is repetitive, we can find a substitution tiling in such that

where , and . The claim follows from Propositions 5.3, 5.7 and 5.8.

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Corollary 5.10   —

Let be a repetitive primitive substitution tiling on with a diagonalizable expansion map ϕ which is unimodular. Suppose that all the eigenvalues of ϕ are algebraically conjugate with the same multiplicity. Then has pure discrete spectrum if and only if each control point set , , is a regular Euclidean model set in CPS (23).

Proof   —

It is known that any regular model sets have pure discrete spectrum in quite a general setting (Schlottmann, 2000 ▸). Together with Theorem 5.9, we obtain the equivalence between pure discrete spectrum and regular model set in substitution tilings.

□

The next example shows that the unimodularity of ϕ is necessary.

Example 5.11   —

Let us consider an example of non-unimodular substitution tiling which is studied by Baake et al. (1998 ▸). This example is proven to be a regular model set in the setting of a CPS constructed by Baake et al. (1998 ▸) with the help of 2-adic embedding. In our setting of CPS (23), we show that this example cannot provide a model set, since we are only interested in the Euclidean window in this paper.

The substitution matrix of the primitive two-letter substitution

has the Perron–Frobenius eigenvalue which is a Pisot number but non-unimodular. We can extend the letter a to the right-hand side by the substitution and the letter b to the left-hand side. So we can get a bi-infinite sequence fixed under the substitution. A geometric substitution tiling arising from this substitution can be obtained by replacing symbols a and b in this sequence by the intervals of length and . Then we have the following tile-substitution ω,

where and . Considering return words for a, and for b, we can check . We choose left end points of corresponding intervals as the set of control points. Then they satisfy

by Lagarias–Wang duality (Lagarias & Wang, 2003 ▸). Applying the Galois conjugate κ which sends , we obtain a generalized iterated function system

with , and . We can easily confirm that

are the unique attractors of this iterated function system. Since contains an inner point, it is unable to distinguish them by any window in this setting.

6. Further study  

We have mainly considered unimodular substitution tilings in this paper. Example 5.11 shows a case of non-unimodular substitution tiling which is studied by Baake et al. (1998 ▸). It cannot be a Euclidean model set in the cut-and-project scheme (23) that we present in this paper, but it is proven to be a regular model set in the setting of a cut-and-project scheme constructed in the work of Baake et al. (1998 ▸), which suggests non-unimodular tilings require non-Archimedean embeddings to construct internal spaces. It is an intriguing open question to construct a concrete cut-and-project scheme in this case.

Funding Statement

This work was funded by National Research Foundation of Korea grant 2019R1I1A3A01060365 to Jeong-Yup Lee. Japan Society for the Promotion of Science grants 17K05159, 17H02849, and BBD30028. Seoul Women‘s University grant 2020-0205 to Dong-il Lee.