## 1. Introduction

The study of long-range aperiodic order has played an important role in understanding the structures of physical models like “quasicrystals”. As an important class of models for the long-range aperiodic order, substitution tilings have been the subject of a great deal of study. Among these substitution tilings, Pisot or Pisot family substitutions get more attention due to their well-ordered properties. A Pisot number is an algebraic integer

$\lambda >1$ for which all of the other algebraic conjugates of

$\lambda $ lie strictly inside the unit circle. If the largest eigenvalue of the corresponding substitution matrix is a Pisot number, we call the substitution a Pisot substitution. There is a well-known conjecture called the “Pisot substitution conjecture”. The associated substitution system always has a pure point spectrum [

1]. For special cases of the Pisot substitutions, the conjecture has been answered affirmatively [

2,

3]. Extending the idea of a Pisot number, one can also look at a Pisot family, a set of algebraic integers whose Galois conjugates with modulus larger than or equal to 1 are all in the same set [

4]. There has been a great deal of of study on Pisot substitution sequences or Pisot family substitution tilings on

${\mathbb{R}}^{d}$ which characterizes the property of pure point spectrum (see [

1] and therein). There are two natural cut-and-project schemes (CPSs) arising in this study. One CPS is constructed with a Euclidean internal space using the Pisot property [

5]. We extend the idea of constructing the CPS from the Pisot property to the Pisot family property. The other CPS is made by constructing an abstract internal space from the property of pure point spectrum [

6]. These two CPSs were developed independently from different aims of study. It is not yet known if these two CPSs have any relation to each other. Here we would like to provide how these two CPSs are related, showing that two internal spaces are basically isomorphic to each other in Theorem 2.

In this article, we consider mainly primitive substitution tilings in

${\mathbb{R}}^{d}$ with pure point spectrum. It is proven in [

7] that primitive substitution tilings with pure point spectrum always show finite local complexity (FLC). So, it is not necessary to make an assumption of FLC in the consideration of pure point spectrum.

In

Section 2, we visit the basic definitions of the terms that we use. In

Section 3, we construct a natural CPS which arises from the property of Pisot family substitution. In

Section 4, we recall the other CPS constructed from the property of pure point spectrum. We show in Theorem 2 that the two CPSs are closely related, demonstrating that there is an isomorphism between two internal spaces of the CPSs under certain model set conditions. In

Section 6, we raise a few questions for later study.

## 2. Preliminary

#### 2.1. Tilings

We consider a set of types (or colours) $\{1,\dots ,\kappa \}$. A tile in ${\mathbb{R}}^{d}$ is a pair $T=(A,i)$ where $A=\mathrm{supp}\left(T\right)$ (the support of T) is a compact set in ${\mathbb{R}}^{d}$ with $A=\overline{{A}^{\circ}}$ and $i=l\left(T\right)\in \{1,\dots ,\kappa \}$ is the type of T. Let $h+T=(h+A,i)$ for $h\in {\mathbb{R}}^{d}$. A set P of tiles is a patch if the number of tiles in P is finite and the interiors of the tiles are mutually disjoint. A tiling $\mathcal{T}$ of ${\mathbb{R}}^{d}$ is a set of tiles for which ${\mathbb{R}}^{d}=\bigcup \{\mathrm{supp}\left(T\right):T\in \mathcal{T}\}$ and the interiors of distinct tiles are disjoint. Given a tiling $\mathcal{T}$, $\mathcal{T}$-patch is a finite set of tiles of $\mathcal{T}$. We always assume that any two $\mathcal{T}$-tiles with the same type are translationally equivalent. Thus, up to translations, there is a finite number of $\mathcal{T}$-tiles.

We will use the following notation:

A van Hove sequence for ${\mathbb{R}}^{d}$ is a sequence $\mathcal{K}={\left\{{K}_{n}\right\}}_{n\ge 1}$ of bounded measurable subsets of ${\mathbb{R}}^{d}$ satisfying

#### 2.2. Delone $\kappa $-Sets

A Delone set in ${\mathbb{R}}^{d}$ is a point set which is relatively dense and uniformly discrete in ${\mathbb{R}}^{d}$. We call $\Lambda ={\left({\Lambda}_{i}\right)}_{i\le \kappa}$ a Delone κ-set in ${\mathbb{R}}^{d}$ if each ${\Lambda}_{i}$ is Delone and $\mathrm{supp}(\Lambda ):={\bigcup}_{i=1}^{\kappa}{\Lambda}_{i}\subset {\mathbb{R}}^{d}$ is Delone. A Delone $\kappa $-set $\Lambda ={\left({\Lambda}_{i}\right)}_{i\le \kappa}$ is called representable if there exist tiles ${T}_{i}=({A}_{i},i),i\le \kappa $, so that $\{x+{T}_{i}:x\in {\Lambda}_{i},i\le \kappa \}$ is a tiling of ${\mathbb{R}}^{d}$.

#### 2.3. Substitutions

**Definition** **1.** Let us consider a finite set $\mathcal{A}=\{{T}_{1},\dots ,{T}_{\kappa}\}$ of tiles in ${\mathbb{R}}^{d}$ with ${T}_{i}=({A}_{i},i)$, $1\le i\le \kappa $ which we will call prototiles.

We denote by ${\mathcal{P}}_{\mathcal{A}}$ the set of patches which are formed by tiles that are translates of ${T}_{i}$’s. $\omega :\mathcal{A}\to {\mathcal{P}}_{\mathcal{A}}$ is a tile-substitution (or substitution) with expansion map ϕ if there are finite sets ${\mathcal{D}}_{ij}\subset {\mathbb{R}}^{d}$ for $i,j\le \kappa $, for whichwithHere all sets in the union have disjoint interiors. It is possible for some of the ${\mathcal{D}}_{ij}$ to be empty. The substitution $\kappa \times \kappa $ matrix $\mathsf{S}$ is defined by $\mathsf{S}(i,j)=\#{\mathcal{D}}_{ij}$. If ${\mathsf{S}}^{m}>0$ for some $m\in \mathbb{N}$, then the corresponding substitution tiling $\mathcal{T}$ is primitive.

A set of algebraic integers $\Theta =\{{\theta}_{1},\cdots ,{\theta}_{r}\}$ is a Pisot family if for any $1\le j\le r$. Every Galois conjugate $\gamma $ of ${\theta}_{j}$, with $\left|\gamma \right|\ge 1$, is contained in $\Theta $. For $r=1$, with ${\theta}_{1}$ real and $|{\theta}_{1}|>1$, this reduces to $|{\theta}_{1}|$ being a real Pisot number, and for $r=2$, with ${\theta}_{1}$ non-real and $|{\theta}_{1}|>1$, to ${\theta}_{1}$ being a complex Pisot number. We say that $\mathcal{T}$ is a Pisot substitution tiling if the expansive map $\varphi :\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$ is a Pisot expansive factor $\lambda $, and a Pisot family substitution tiling if the eigenvalues of the expansive map $\varphi :\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$ form a Pisot family.

#### 2.4. Cut-and-Project Scheme

**Definition** **2.** A cut and project scheme (CPS) is a collection of spaces and mappings as follows:

where

${\mathbb{R}}^{d}$ is a real Euclidean space,

H is a locally compact Abelian group,

${\pi}_{1}$ and

${\pi}_{2}$ are the canonical projections,

$\tilde{L}$ is a lattice in

${\mathbb{R}}^{d}\times H$ (i.e., a discrete subgroup such that the quotient group

$({\mathbb{R}}^{d}\times H)/\tilde{L}$ is compact),

${\pi}_{1}{|}_{\tilde{L}}$ is injective, and

${\pi}_{2}(\tilde{L})$ is dense in

H.

For a subset $V\subset H$, we denote $\Lambda \left(V\right):=\{{\pi}_{1}\left(x\right)\in {\mathbb{R}}^{d}:x\in \tilde{L},{\pi}_{2}\left(x\right)\in V\}$.

We define a model set in ${\mathbb{R}}^{d}$ as a subset $\Lambda $ in ${\mathbb{R}}^{d}$ for which, up to translation, $\Lambda \left({W}^{\circ}\right)\subset \Gamma \subset \Lambda \left(W\right)$, W is compact in H, $W=\overline{{W}^{\circ}}\ne \varnothing $. If the boundary $\partial W=W\backslash {W}^{\circ}$ of W is of (Haar) measure 0, we say that the model set $\Lambda $ is regular. We say that $\Lambda ={\left({\Lambda}_{i}\right)}_{i\le \kappa}$ is a model $\kappa $-set (resp. regular model $\kappa $-set) if each ${\Lambda}_{i}$ is a model set (resp. regular model set) with regard to the same CPS.

We assume, without loss of generality, that

H is generated by the windows

${W}_{i}$’s, where

${\Lambda}_{i}=\Lambda \left({W}_{i}\right)$ for all

$i\le \kappa $. When

H satisfies the following:

we say that the windows

${W}_{i}$’s have irredundancy.

#### 2.5. Pure Point Spectrum

Let

${X}_{\mathcal{T}}$ be the set of all primitive substitution tilings in

${\mathbb{R}}^{d}$ such that clusters of each tiling are translates of a

$\mathcal{T}$-patch. We give a usual metric

$\delta $ in tilings in such a way that two tilings are close if there is a large agreement on a big area with small shift (see [

8,

9,

10]). Then,

${X}_{\mathcal{T}}=\overline{\{-h+\mathcal{T}:h\in {\mathbb{R}}^{d}\}}$, where we take the closure in the topology induced by the metric

$\delta $. We consider a natural action of

${\mathbb{R}}^{d}$ by translations on the dynamical hull

${X}_{\mathcal{T}}$ of

$\mathcal{T}$ and get a topological dynamical system

$({X}_{\mathcal{T}},{\mathbb{R}}^{d})$. Assume that

$({X}_{\mathcal{T}},\mu ,{\mathbb{R}}^{d})$ is a measure-preserving dynamical system with a unique ergodic measure

$\mu $. We look at the associated group of unitary operators

${\left\{{T}_{x}\right\}}_{x\in {\mathbb{R}}^{d}}$ on

${L}^{2}({X}_{\mathcal{T}},\mu )$:

Every $g\in {L}^{2}({X}_{\mathcal{T}},\mu )$ defines a function on ${\mathbb{R}}^{d}$ by $x\mapsto \langle {T}_{x}g,g\rangle $, which is positive definite on ${\mathbb{R}}^{d}$. So, its Fourier transform is a positive measure ${\sigma}_{g}$ on ${\mathbb{R}}^{d}$ and we call it the spectral measure corresponding to g. We say that the dynamical system $({X}_{\mathcal{T}},\mu ,{\mathbb{R}}^{d})$ has pure point spectrum if ${\sigma}_{g}$ is pure point for each $g\in {L}^{2}({X}_{\mathcal{T}},\mu )$. If the dynamical system $({X}_{\mathcal{T}},\mu ,{\mathbb{R}}^{d})$ has a pure point spectrum, we also say that $\mathcal{T}$ has a pure point spectrum.

## 3. Cut-and-Project Scheme for Pisot Family Substitution Tilings

We consider a primitive substitution tiling $\mathcal{T}$ on ${\mathbb{R}}^{d}$ with expansion map $\varphi $. There is a standard way to choose distinguished points in the tiles of primitive substitution tiling so that they form a $\varphi $-invariant Delone set. They are called control points. A tiling $\mathcal{T}$ is called a fixed point of the substitution $\omega $ if $\omega \left(\mathcal{T}\right)=\mathcal{T}$.

**Definition** **3** ([

11,

12])

**.** Let $\mathcal{T}$ be a fixed point of a primitive substitution with expansion map ϕ. For every $\mathcal{T}$-tile T, we choose a tile $\gamma T$ on the patch $\omega \left(T\right)$. For all tiles of the same type, we choose $\gamma T$ with the same relative position. This defines a map $\gamma :\mathcal{T}\to \mathcal{T}$ called the tile map. Then, we define the control point for a tile $T\in \mathcal{T}$ byThe control points satisfy the following:

- (a)
${T}^{\prime}=T+c\left({T}^{\prime}\right)-c\left(T\right)$, for any tiles $T,{T}^{\prime}$ of the same type;

- (b)
$\varphi \left(c\right(T\left)\right)=c\left(\gamma T\right)$, for $T\in \mathcal{T}$.

For tiles of any tiling

$\mathcal{S}\in {X}_{\mathcal{T}}$, control points have the same relative position as in

$\mathcal{T}$-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

be a set of control points of the tiling

$\mathcal{T}$ in

${\mathbb{R}}^{d}$. Let

where

${\mathcal{C}}_{i}$ is the set of control points of tiles of type

i.

Let us assume that

$\varphi $ is diagonalizable over

$\mathbb{C}$ and the eigenvalues of

$\varphi $ are algebraically conjugate with multiplicity one. For a complex eigenvalue

$\lambda $ of

$\varphi $, the

$2\times 2$ diagonal block

$\left[\begin{array}{cc}\lambda \hfill & 0\hfill \\ 0\hfill & \overline{\lambda}\hfill \end{array}\right]$ is similar to a real

$2\times 2$ matrix

where

$\lambda =a+ib$,

$a,b\in \mathbb{R}$, and

$S=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}\hfill 1& \hfill i\\ \hfill 1& \hfill -i\end{array}\right]$. So we can assume, by appropriate choice of basis, that

$\varphi $ is diagonal with the diagonal entries equal to

$\lambda $ corresponding to real eigenvalues, and diagonal

$2\times 2$ blocks of the form

$\left[\begin{array}{cc}\hfill {a}_{j}& \hfill -{b}_{j}\\ \hfill {b}_{j}& \hfill {a}_{j}\end{array}\right]$ corresponding to complex eigenvalues

${a}_{j}+i{b}_{j}$.

We assume, without loss of generality, that $\varphi $ is a diagonal matrix.

We recall the following theorem. The theorem is not in the form as shown here, but one can readily note that from the proof of [

4] (Theorem 4.1).

**Theorem** **1** ([

4] Theorem 4.1)

**.** Let $\mathcal{T}$ be a primitive substitution tiling on ${\mathbb{R}}^{d}$ with expansion map ϕ. We assume that $\mathcal{T}$ has FLC, ϕ is diagonalizable, and all the eigenvalues of ϕ are algebraically conjugate with multiplicity one. Then, there exists an isomorphism $\sigma :{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$ such thatwhere $\alpha =(1,1,\cdots ,1)\in {\mathbb{R}}^{d}$ and $k\in \mathbb{Z}$.

Let us assume now that

$\mathcal{T}$ has FLC,

$\varphi $ is diagonalizable, the eigenvalues of

$\varphi $ are all algebraically conjugate with multiplicity one, and there exists at least one other algebraic conjugate different from eigenvalues of

$\varphi $. Suppose that

$\varphi $ has

e number of real eigenvalues, and

f number of

$2\times 2$ blocks of the form of complex eigenvalues where

$d=e+2f$. Let all the algebraic conjugates of eigenvalues of

$\varphi $ be real numbers

${\lambda}_{1},\dots ,{\lambda}_{s}$ and complex numbers

${\lambda}_{s+1},{\overline{\lambda}}_{s+1},\dots ,{\lambda}_{s+t},{\overline{\lambda}}_{s+t}$. Let

$m:=s+2t$ and write

${\lambda}_{s+t+i}={\overline{\lambda}}_{s+i}$ for

$i=1,\cdots t$ for convenience. Let us consider a space

$\mathbb{K}$, where

Let us consider the following map:

where

$P\left(x\right)$ is a polynomial over

$\mathbb{Z}$. Let us build a new cut and project scheme:

where

${\pi}_{1}$ and

${\pi}_{2}$ are canonical projections,

$L={\langle {\mathcal{C}}_{i}\rangle}_{i\le \kappa}$, and

$\tilde{L}=\{(x,\Psi \left(x\right)):x\in L\}$. It is clear to see that

${\pi}_{1}{|}_{\tilde{L}}$ is injective. We now show that

${\pi}_{2}\left(\tilde{L}\right)$ is dense in

$\mathbb{K}$ and

$\tilde{L}$ is a lattice in

${\mathbb{R}}^{d}\times \mathbb{K}$.

**Lemma** **1.** $\tilde{L}$ is a lattice in ${\mathbb{R}}^{d}\times \mathbb{K}$.

**Proof.** Since $\mathbb{Z}\left[\varphi \right]\alpha $ is a free $\mathbb{Z}$-module of rank m and $m\times m$ matrix $A={\left({\lambda}_{i}^{j-1}\right)}_{i,j\in \{1,\cdots ,m\}}$ is non-degenerate by the Vandermonde determinant, the natural embedding combining all conjugates; $f:\mathbb{Z}\left[\varphi \right]\alpha \to {\mathbb{R}}^{s}\times {\mathbb{C}}^{t}\simeq {\mathbb{R}}^{d}\times \mathbb{K}$ gives a lattice $f\left(\mathbb{Z}\right[\varphi \left]\alpha \right)$ in ${\mathbb{R}}^{d}\times \mathbb{K}$. Consequently, $\tilde{L}$ is isomorphic to a free $\mathbb{Z}$-submodule of $f\left(\mathbb{Z}\right[\varphi \left]\alpha \right)$ due to the theory of elementary divisors. From Theorem 1, $\tilde{L}$ is isomorphic to a full rank $\mathbb{Z}$-submodule of $f\left(\mathbb{Z}\right[\varphi \left]\alpha \right)$, that is, a sub-lattice of $f\left(\mathbb{Z}\right[\varphi \left]\alpha \right)$. Thus, the claim is shown. The case with complex conjugates can be shown in a similar manner, taking care of embeddings $\mathbb{C}$ to ${\mathbb{R}}^{2}$. □

**Lemma** **2.** $\Psi \left(L\right)={\pi}_{2}(\tilde{L})$ is dense in $\mathbb{K}$.

**Proof.** We showed that

$\tilde{L}$ is a sub-lattice of

$f\left(\mathbb{Z}\right[\varphi \left]\alpha \right)$ in the proof of Lemma 1. So it suffices to prove that

$\Psi \left(\mathbb{Z}\right[\varphi \left]\right)$ is dense in

$\mathbb{K}$. We prove the totally real case, that is,

${\lambda}_{i}\in \mathbb{R}$ for all

i. By [

13] (Theorem 24),

$\Psi \left(\mathbb{Z}\right[\varphi \left]\right)$ is dense if

implies

${x}_{i}=0$ for

$i=d+1,\cdots ,m$. The condition is equivalent to

with

$\xi =\left({x}_{i}\right)=(0,\cdots ,0,{x}_{d+1},\cdots ,{x}_{m})\in {\mathbb{R}}^{m}$ in the terminology of Lemma 1. Multiplying the inverse of

A, we see that entries of

$\xi $ must be Galois conjugates. As

$\xi $ has at least one zero entry, we obtain

$\xi =0$, which shows

${x}_{i}=0$ for

$i=d+1,\cdots ,m$. In fact, this discussion uses the Pontryagin duality that the

$\Psi :{\mathbb{Z}}^{m}\to {\mathbb{R}}^{m-d}$ has a dense image if and only if its dual map

$\widehat{\Psi}:{\mathbb{R}}^{m-d}\to {\mathbb{T}}^{m}$ is injective (see also [

14,

15,

16] [Chapter II,

Section 1]). The case with complex conjugates is similar. □