**Cut-and-Project Schemes for Pisot Family**

**Substitution Tilings**

**Jeong-Yup Lee1,*ID _{, Shigeki Akiyama}2_{and Yasushi Nagai}3**

1 _{Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701,}

Korea and KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea; jylee@cku.ac.kr

2 _{Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, Japan (zip:305-8571);}

akiyama@math.tsukuba.ac.jp

3 _{Montanuniversität, Department Mathematik und Informationstechnologie, Lehrstuhl für Mathematik und}

Statistik, Franz Josef Strasse 18, A-8700 Leoben, Austria; yasushi.nagai@unileoben.ac.at

***** Correspondence: jylee@cku.ac.kr; Tel.: +82-10-7726-8606
Version September 19, 2018 submitted to Preprints

**Abstract:** We consider Pisot family substitution tilings inRdwhose dynamical spectrum is pure
1

point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family

2

property and the other from the pure point spectrum respectively. The first CPS has an internal

3

spaceRmfor some integerm∈Ndefined from the Pisot family property, and the second CPS has 4

an internal spaceHwhich is an abstract space defined from the property of the pure point spectrum.

5

However it is not known how these two CPS’s are related. Here we provide a sufficient condition

6

to make a connection between the two CPS’s. In the case of Pisot unimodular substitution tiling in

7

R, the two CPS’s turn out to be same due to [5, Remark 18.5]. 8

**Keywords:**Pisot substitution tilings; pure point spectrum; regular model set; algebraic coincidence

9

**0. Introduction**

10

There has been a lot of study on Pisot substitution sequences or Pisot family substitution tilings

11

onRdwhich characterizes the property of pure point spectrum (see [2] and therein). There are two 12

natural cut-and-project schemes(CPS) arising in this study. One CPS is constructed with an Euclidean

13

internal space using the Pisot property [5]. We extend the idea of constructing the CPS from the Pisot

14

property to the Pisot family property. The other CPS is made by constructing an abstract internal

15

space from the property of pure point spectrum [4]. These two CPS’s got developed independently

16

from different aims of study. It is not known yet if these two CPS’s have any relation to each other.

17

Here we would like to provide how these two CPS’s are related showing that two internal spaces are

18

basically isomorphic to each other in Theorem3.1.

19

In this paper we are mainly interested in primitive substitution tilings inRdwith pure point 20

spectrum. It is shown in [17] that primitive substitution tilings with pure point spectrum always

21

have finite local complexity (FLC). So it is not necessary to make an assumption of FLC in the

22

consideration of pure point spectrum.

23

In section1, we visit the basic definitions of the terms that we use. In section2, we construct a

24

natural CPS which arises from the property of Pisot family substitution. In section3, we recall the

25

other CPS constructed from the property of pure point spectrum. We show in Theorem3.1that the

26

two CPS’s are closely related showing that there is an isomorphism between two internal spaces of

27

the CPS’s under certain model set condition. In section4, we raise a few questions for later study.

28

**1. Preliminary**

29

1.1. Tilings

30

We begin with a set of types (or colours){1, . . . ,*κ}*, which we fix once and for all. AtileinRd
31

is defined as a pairT= (A,i)whereA=supp(T)(the support ofT) is a compact set inRd, which 32

is the closure of its interior, andi=l(T)∈ {1, . . . ,*κ}*is the type ofT. We letg+T= (g+A,i)for

33

g∈Rd. We say that a setPof tiles is apatchif the number of tiles inPis finite and the tiles ofPhave 34

mutually disjoint interiors. A tiling ofRdis a setT of tiles such thatRd=S{supp(T):T∈ T }and 35

distinct tiles have disjoint interiors. Given a tilingT, a finite set of tiles ofT is calledT-patch. We

36

always assume that any twoT-tiles with the same colour are translationally equivalent (hence there

37

are finitely manyT-tiles up to translations).

38

We will make use of the following notation:

39

F+r:={x ∈_{R}d: dist(x,F)≤r} and F−r :={x ∈F: dist(x,*∂F*)≥r}. (1.1)

Avan Hove sequenceforRdis a sequenceF ={Fn}n≥1of bounded measurable subsets ofRd 40

satisfying

41

lim

n→∞Vol((*∂Fn*)

+r_{)/Vol(}_{Fn}_{) =}_{0, for all}_{r}_{>}_{0.} _{(1.2)}

1.2. Delone*κ-sets*

42

Recall that a Delone set is a relatively dense and uniformly discrete subset inRd. We say that 43

**Λ**= (Λi)i≤*κ*is aDelone*κ-set*inRdif eachΛiis Delone and supp(**Λ**):=
S_{κ}

i=1Λi ⊂Rdis Delone.

44

A Delone*κ*-set**Λ**= (Λi)i≤*κ* is calledrepresentable(by tiles) if there exist tilesTi= (Ai,i),i≤*κ*,
45

so that{x+Ti:x∈Λi,i≤*κ}*is a tiling ofRd, that is,Rd=Si≤*κ*
S

x∈Λi(x+Ai), and the sets in this

46

union have disjoint interiors.

47

1.3. Substitutions

48

Definition1. LetA={T_{1}, . . . ,T*κ*}be a finite set of tiles inRdsuch thatTi= (Ai,i); we will call them
prototiles. Denote byPAthe set of patches made of tiles each of which is a translate of one ofTi’s. We
say that*ω*:A → PAis atile-substitution(or simplysubstitution) with expansive map*φ*if there exist
finite setsD_{ij}⊂_{R}d_{for}_{i}_{,}_{j}_{≤}

*κ*, such that

*ω*(Tj) ={u+Ti: u∈ Dij, i=1, . . . ,*κ}* (1.3)

with

49

*φA*j =

*κ*
[

i=1

(D_{ij}+Ai) forj≤*κ*. (1.4)

Here all sets in the right-hand side must have disjoint interiors; it is possible for some of theDijto be

50

empty. Thesubstitution*κ*×*κ*matrixSis defined byS(i,j) =#Dij. IfSm>0 for somem∈N, we say
51

that the substitution tilingT isprimitive.

A set of algebraic integersΘ={θ_{1},· · ·,*θr*}is aPisot familyif for any 1≤ j≤r, every Galois

53

conjugate*γ*of*θ*j, with|γ| ≥1, is contained inΘ. Forr=1, with*θ*1real and|θ1|>1, this reduces to

54

|θ1|being a real Pisot number, and forr=2, with*θ*1non-real and|θ1| >1, to*θ*1being a complex

55

Pisot number. We say thatT is aPisot substitution tilingif the expansive map*φ* : Rd → Rd is a
56

Pisot expansive factor*λ*, and aPisot family substitution tilingif the eigenvalues of the expansive map

57

*φ*: Rd→Rdform a Pisot family.
58

1.4. Cut and project scheme

59

Definition2. Acut and project scheme(CPS) consists of a collection of spaces and mappings as follows;

60

Rd ←−*π*1 Rd×H −→*π*2 H
S

e

L

(1.5)

whereRdis a real Euclidean space,His a locally compact Abelian group,*π*1and*π*2are the canonical

61

projections,eL⊂Rd×His a lattice, i.e. a discrete subgroup for which the quotient group(Rd×H)/eL 62

is compact,*π*1|eLis injective, and*π*2(eL)is dense inH.

63

For a subsetV⊂H, we denoteΛ(V):={π1(x)∈Rd:x∈eL,*π*_{2}(x)∈V}.
64

Amodel setinRdis a subsetΛofRdfor which, up to translation,Λ(W◦) ⊂Γ ⊂Λ(W),Wis 65

compact inH,W =W◦ _{6}_{=}_{∅}_{. The model set}_{Λ}_{is}_{regular}_{if the boundary}_{∂W}_{=}_{W\W}◦_{of}_{W}_{is of}

66

(Haar) measure 0. We say that**Λ**= (Λi)i≤*κ*is amodel*κ-set*(resp.regular model*κ-set*) if eachΛiis a
67

model set (resp. regular model set) with respect to the same CPS.

68

Without loss of generality, we assume thatHis generated by the windowsWi’s, whereΛi =
Λ(Wi)for alli≤*κ*. WhenHsatisfies the following

{t∈ H:t+Wi=Wifor alli≤*κ}*={0}, (1.6)

we say that the windowsWi’s haveirredundancy.

69

1.5. Pure point spectrum

70

LetXT be the collection of all primitive substitution tilings inRdeach of whose clusters is
a translate of a T-patch. We give a usual metric *δ*on tilings in such a way that two tilings are
close if there is a large agreement on a big area with small shift (see [15,24,26]). Then XT =
{−h+T :h∈Rd}where the closure is taken in the topology induced by the metric*δ*. We have a
natural action ofRdon the dynamical hullXT ofT by translations and get a topological dynamical
system(XT,Rd). Let(XT,*µ*,Rd)be a measure preserving dynamical system with a unique ergodic
measure*µ*. We consider the associated group of unitary operators{Tx}_{x∈}_{R}d onL2(XT,*µ*):

Txg(T0) =g(−x+T0).

Everyg∈L2(XT,*µ*)defines a function onRdbyx7→ hTxg,gi. This function is positive definite on
71

Rd, so its Fourier transform is a positive measure*σg*onRdcalled thespectral measurecorresponding
72

tog. The dynamical system(XT,*µ*,Rd)is said to havepure point spectrumif*σg*is pure point for every

g∈L2(XT,*µ*). We also say thatT has pure point spectrum if the dynamical system(XT,*µ*,Rd)has
74

pure point spectrum.

75

**2. Cut-and-project scheme for Pisot family substitution tilings**

76

LetT be a primitive substitution tiling onRdwith expansion map*φ*. There is a standard way to

77

choose distinguished points in the tiles of primitive substitution tiling so that they form a*φ*-invariant

78

Delone set. They are calledcontrol points. A tilingT is called a fixed point of the substitution*ω*if

79

*ω*(T) =T.

80

Definition3 ([23,28]). LetT be a fixed point of a primitive substitution with expansion map*φ. For each*
T-tile T, fix a tile*γT in the patchω*(T); choose*γT with the same relative position for all tiles of the same*
type. This defines a map*γ*:T → T called the tile map. Then define the control point for a tile T∈ T by

{c(T)}= ∞ \

n=0

*φ*−n(*γ*nT).

The control points have the following properties:

81

(a) T0=T+c(T0)−c(T), for any tilesT,T0of the same type;

82

(b) *φ*(c(T)) =c(*γT*), forT∈ T.

83

Control points are also fixed for tiles of any tilingS ∈ XT: they have the same relative position as inT-tiles. Note that the choice of control points is non-unique, but there are only finitely many possibilities, determined by the choice of the tile map. Let

C:=C(T) ={c(T):T∈ T }

be a set of control points of the tilingT inRd. Let

Ξ:=Ξ(T) =

*κ*
[

i=1

(Ci− Ci)

whereCiis the set of control points of tiles of typei. Equivalently,Ξis the set of translation vectors

84

between twoT-tiles of the same type.

85

Let us assume that *φ* is diagonalizable over C and the eigenvalues of *φ* are algebraically

86

conjugate with multiplicity one. For a complex eigenvalue*λ*of*φ*, the 2×2 diagonal block

"

*λ* 0

0 *λ*

#

87

is similar to a real 2×2 matrix

88

"

a −b

b a

#

=S−1

"

*λ* 0

0 *λ*

#

S, (2.1)

where*λ* = a+ib,a,b∈ R, andS = √1_{2}

"

1 i 1 −i

#

. So we can assume, by appropriate choice of

89

basis, that*φ*is diagonal with the diagonal entries equal to*λ*corresponding to real eigenvalues, and

90

diagonal 2×2 blocks of the form

"

aj −bj bj aj

#

corresponding to complex eigenvaluesaj+ibj.

Without loss of generality, we can assume that*φ*is a diagonal matrix.

92

We recall the following theorem. The theorem is not in the form as shown here, but one can

93

readily note that from the proof of [16, Thm.4.1].

94

**Theorem 2.1.** [16, Thm.4.1] LetT be a primitive substitution tiling onRdwith expansion map*φ. Assume*
thatT has FLC,*φ*is diagonalizable, and all the eigenvalues of*φ*are algebraically conjugate with multiplicity
one. Then there exists an isomorphism*σ*:Rd→Rdsuch that

*σφ*=*φσ* and k·Z[*φ*]*α*⊂*σ*(C(T))⊂Z[*φ*]*α*,

where*α*= (1, 1, . . . , 1)∈Rdand k∈Z.
95

Let us assume now thatT has FLC,*φ*is diagonalizable, the eigenvalues of*φ*are all algebraically
conjugate with multiplicity one, and there exists at least one other algebraic conjugate different from
eigenvalues of*φ*. Suppose that*φ*hasenumber of real eigenvalues, and f number of 2×2 blocks of
the form of complex eigenvalues whered=e+2f. Let all the algebraic conjugates of eigenvalues of
*φ*be real numbers*λ*1, . . . ,*λs*and complex numbers*λ*s+1,*λ*s+1, . . . ,*λs*+t,*λs*+t. Letm:=s+2tand
write*λ*s+t+i =*λ*s+ifori=1, . . .tfor the convenience. Let us consider a spaceK, where

K:=Rs−e×Ct−f wRm−d.

Let us consider a following map

96

Ψ:Z[*φ*]*ξ*→K, (2.2)

P(*φ*)*ξ*7→(P(*λe*+1), . . . ,P(*λs*),P(*λ*s+f+1), . . . ,P(*λs*+t)), (2.3)

whereP(x)is a polynomial overZ. Let us construct a new cut and project scheme : 97

Rd ←−*π*1 Rd×K −→*π*2 K
∪

L ←− eL −→ Ψ(L)

x ←−[ (x,Ψ(x)) 7−→ Ψ(x),

(2.4)

where*π*1and*π*2are canonical projections,L=hCiii≤*κ* andeL={(x,Ψ(x)):x∈ L}. It is clear to see
98

that*π*1|eLis injective. We now show that*π*2(eL)is dense inKandeLis a lattice inR
d_{×}

K. 99

**Lemma 4.** eL is a lattice inRd×K.
100

**Proof.** Since Z[*φ*]*α* is a free Z-module of rank m and m×m matrix A = (*λ*j−_{i} 1)_{i}_{,}j∈{1,...,m} is

101

non-degenerate by the Vandermonde determinant, the natural embedding combining all conjugates;

102

f :Z[*φ*]*α*→Rs×Ct 'Rd×Kgives a lattice f(Z[*φ*]*α*)inRd×K. ConsequentlyeLis isomorphic
103

to a freeZ-submodule of f(Z[*φ*]*α*)due to the theory of elementary divisors. From Theorem2.1,

104

claim is shown. The case with complex conjugates can be shown in a similar manner, taking care of

106

embeddingsCtoR2. 107

**Lemma 5.** Ψ(L) =*π*2(eL)is dense inK.
108

**Proof.** As in the proof of Lemma4, we showed thateLis a sub-lattice of f(Z[*φ*]*α*), it suffices to prove

thatΨ(_{Z}[*φ*])is dense inK. We prove the totally real case, i.e.,*λ*i∈Rfor alli. By [27, Theorem 24],
Ψ(Z[*φ*])is dense if

m

### ∑

i=d+1xi*λ*_{i}j−1∈Z (j=1, . . . ,m)

impliesxi=0 fori=d+1, . . . ,m. The condition is equivalent to

*ξ*A∈Zm

with*ξ* = (xi) = (0, . . . , 0,x_{d}+1, . . . ,xm) ∈ Rmin the terminology of Lemma 4. Multiplying the

109

inverse ofA, we see that entries of*ξ*must be Galois conjugates. As*ξ*has at least one zero entry, we

110

obtain*ξ*=0 which showsxi =0 fori=d+1, . . . ,m. In fact, this discussion is using the Pontryagin

111

duality that theΨ :Zm → Rm−dhas a dense image if and only if its dual mapΨb : Rm−d → Tm 112

is injective (see also [18, Chapter II, Section 1], [10] and [1]). The case with complex conjugates is

113

similar.

114

**3. Two cut-and-project schemes**

115

3.1.*φ-topology*

116

LetT be a primitive substitution tiling onRdwith expansion map*φ*. Define

L:=hC_{i}ii≤*κ*

be the group generated byCi,i≤*κ*, whereC = (Ci)i≤*κ*is a control point set ofT and

K:={x∈Rd:T +x=T }

be the set of periods ofT. We say thatT admits analgebraic coincidenceif there existM∈_{Z}+and
117

*ξ*∈ Cifor somei≤*κ*such that*ξ*+QMΞ(T)⊂ Ci. It is known in [11] thatT has pure point spectrum

118

if and only ifT admits an algebraic coincidence.

119

Under the assumption thatT admits an algebraic coincidence, we introduce a topology onLand

120

find a completionHof the topological groupLsuch that the image ofLis a dense subgroup ofH. This

121

enables us to construct a cut and project scheme (CPS) such that each point setCi,i≤*κ*, arises from

122

the CPS. From the following lemma, we know that the system{α+*φ*nΞ(T) +K:n∈Z+,*α*∈L}

123

satisfies the topological properties for the groupLto be a topological group ([19], [7] and [20]).

124

**Lemma 6.** [11, Lemma 4.1] LetT be a primitive substitution tiling with an expansive map*φ. Suppose that*

125

T admits an algebraic coincidence. Then the system{φn_{Ξ}_{(}_{T}_{) +}_{K}_{:}_{n}_{∈}

Z+}serves as a neighbourhood 126

base for0∈ L of the topology on L relative to which L becomes a topological group.

We call the topology on Lwith the neighbourhood base{α+*φ*nΞ(T) +K :n ∈ Z+,*α* ∈ L}

128

*φ-topology*. LetL*φ*be the spaceLwith*φ*-topology.
129

LetL0 = L/K. From [7, III. §3.4, §3.5] and Lemma6, we know that there exists a complete

130

Hausdorff topological group ofL0, which we denote by H, for whichL0is isomorphic to a dense

131

subgroup of the complete groupH(see [4] and [14]). Furthermore there is a uniformly continuous

132

mapping*ψ*:L→Hwhich is the composition of the canonical injection ofL0intoHand the canonical

133

homomorphism ofLontoL0for which*ψ*(L)is dense inHand the mapping*ψ*fromLonto*ψ*(L)is

134

an open map, the latter with the induced topology of the completionH. One can directly considerH

135

as the Hausdorff completion ofLvanishingK.

136

3.2. P*e*-topology
137

We introduce another topology on L which becomes equivalent to *φ*-topology under the

138

assumption of algebraic coincidence.

139

Let{Fn}n∈Z+ be a van Hove sequence and letT

0_{,}_{T}00_{be two tilings in}

Rd, where**Λ**0 = (Λ0i)i≤*κ*

and**Λ**00= (Λ00_{i})i≤*κ*are representable Delone*κ*-sets of the tilingsT

0_{,}_{T}00_{. We define}

*ρ*(T0,T00):= lim
n→∞sup

∑*κ*

i=1]((Λ0i4Λ00i)∩Fn)

Vol(Fn) . (3.1)

Here4is the symmetric difference operator. LetP*e* ={x ∈ L: *ρ*(x+T,T)<*e}*for each*e* >0.
140

IfT admits an algebraic coincidence, then, for any*e*>0,P*e*is relatively dense [3,4,15,22]. In this
141

case the system{P*e*:*e*>0}serves as a neighbourhood base for 0∈Lof the topology onLrelative
142

to whichLbecomes a topological group. We nameP*e*-topology for this topology onLand denote

143

the spaceLwithP*e*-topology byLP(see [3,4] forP*e*-topology under the name of autocorrelation

144

topology).

145

**Proposition 7.** [11, Prop. 4.6, 4.7] LetT be a primitive substitution tiling. Suppose thatT admits an

146

algebraic coincidence, then the mapping*ι*:x 7→x from L*φ*onto LPis topologically isomorphic.
147

**Remark 8.** From Prop.7, LP is topologically isomorphic to L*φ*. Thus the completion of LP is

148

topologically isomorphic to the completion Hof L*φ*. We will identify the former withH. Thus
149

*ϕ*:= *ψ*·*ι*−1: LP→ His uniformly continuous,*ϕ*(LP)is dense inH, and the mapping *ϕ*fromLP

150

onto*ϕ*(LP)is an open map, the latter with the induced topology of the completionH. Therefore we

151

can consider the CPS (1.5) with an internal spaceHwhich is a completion ofLP. Note that sinceT is

152

repetitive,T

*e*>0P*e*=KandK={0}inL*φ*.
153

We observe thatLPandΨ(L)are all topologically isomorphic when the control point setCis a

154

regular model*κ*-set in CPS(2.4).

155

**Theorem 3.1.** LetT be a primitive Pisot family substitution tiling inRdwith an expansive map*φ. Suppose*
156

that*φ*is diagonalizable, all the eigenvalues of*φ*are algebraic conjugates with multiplicity one, and there

157

exists at least one algebraic conjugate*λ*of eigenvalues of*φ*for which|λ|<1. IfCis a regular model*κ-set*

158

in CPS(2.4), then the internal space H which is the completion of L*φ*with*φ-topology is isomorphic to the*
159

PROOF. Since*φ*is an expansive map and satisfies the Pisot family condition, we first note that there

161

is no algebraic conjugate*γ*of eigenvalues of*φ*with|γ|=1.

162

We will show that if fort∈L,Ψ(t)is close to 0 inK, then*ρ*(t+T,T)is close to 0 inH. Since

163

each point setC_{i}is a regular model set by the assumption whereC= (C_{i})i≤*κ* andCi =Λ(Wi)in the

164

CPS (2.4), fort∈L

165

*ρ*(t+T,T) = lim
n→∞sup

∑*κ*

i=1](((t+Ci)4 Ci)∩An) Vol(An)

=

*κ*

### ∑

i=1lim n→∞

](((t+Ci)4 Ci)∩An) Vol(An)

=

*κ*

### ∑

i=1(*θ*(Wi\(Ψ(t) +Wi)) +*θ*(Wi\(−Ψ(t) +Wi))), (3.2)

where*θ*is a Haar measure inK(see [21, Thm. 1]).
166

Note that

*θ*(Wi\(s+Wi)) =*θ*(Wi)−**1W**i∗g**1W**i(s)

is uniformly continuous ins ∈ _{K} (see [25, Subsec. 1.1.6]). So ifΨ(t)converges to 0 inK, then
167

*ρ*(t+T,T)converges to 0 inR.
168

On the other hand, suppose that{tn}is a sequence such that*ρ*(tn+T,T)→0 asn→∞. Then
for eachi≤*κ*

{θ(Wi\(Ψ(tn) +Wi))}n→0 asn→∞.

Note that for large enoughn,Wi∩(Ψ(tn) +Wi) 6= ∅and soΨ(tn) ∈Wi−Wifor alli≤ *κ*. Since
Wi−Wi is compact,{Ψ(tn)}n has a converging subsequence{Ψ(tn_{k})}k. For any such sequence
definet0∗:=limk→∞Ψ(tnk). Then

*θ*(Wi\(t0∗+Wi)) =0

and so*θ*(Wi◦\(t0∗+Wi)) =0 for eachi≤*κ*. ThusWi◦⊂t0∗+Wiand this impliesWi⊂t0∗+Wi.

169

On the other hand, limk→∞−Ψ(tnk) = −t0

∗ _{and}

*θ*(Wi◦\(−t0∗+Wi)) = 0. SoWi ⊂ −t0∗+Wi.

170

HenceWi ⊂ t0∗+Wi ⊂ t0∗−t0∗+Wi andWi = t0∗+Wi. This equality is for eachi ≤ *κ*. Since

171

Kis isomorphic toRm−d, each model setWihas irredundancy. Thust0∗ = 0. So all converging

172

subsequences{Ψ(tn_{k})}kconverge to 0 and{Ψ(tn)}n →0 asn→∞.

173

This establishes the equivalence of the two topologies. By [7, Prop 5, III §3.3], there exists an

174

isomorphism betweenHontoK. 175

The above theorem shows that the internal spaceHconstructed fromL*φ*with*φ*-topology is
176

isomorphic to Euclidean spaceK(i.e.Rm−d). 177

It is known in [5,9], [2, Thm. 3.6] that unimodular irreducible Pisot substitution tilings inRwith 178

pure point spectrum give rise to regular model sets. We give a precise statement below.

179

**Theorem 3.2.** [5, Remark 18.5] LetT be a primitive substitution tiling inRwith expansion factor*β*being a

180

unimodular irreducible Pisot number. ThenT has pure point spectrum if and only if for any1≤i≤*κ, each*

181

C_{i}is a regular model set in CPS (2.4).

**Corollary 9.** LetT be a primitive Pisot substitution tiling inRwith an expansion factor*β. Assume that*
183

there exists at least one algebraic conjugate*λ*of*β*for which|λ| < 1. IfT has pure point spectrum, then

184

the internal space H which is the completion of L*φ*with*φ-topology can be realized by Euclidean space*Rm−1
185

where m is the degree of the characteristic polynomial of*β.*

186

PROOF. By Theorem3.2, it is known that for a primitive Pisot substitution tiling inR, ifT has pure 187

point spectrum, thenCis a regular model*κ*-set in CPS(2.4).

188

**4. Further study**

189

We are left with the following questions extending Theorem3.1.

190

**Question 10.** Can we replace the assumption of regular model*κ-set by pure point spectrum? Another words,*

191

for a primitive Pisot family substitution tilingT inRdwith an expansion map*φ, does the pure point spectrum*
192

ofT imply thatCis a regular model*κ-set with an Euclidean internal space.*

193

**Question 11.** Can the theorem be still extended into the case that the multiplicity of eigenvalues of*φ*is not

194

one?

195

**Author Contributions:**Writing – original draft, Jeong-Yup Lee, Shigeki Akiyama and Yasushi Nagai

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**Acknowledgments:** The first author would like to acknowledge the support by Local University Excellent

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Researcher Supporting Project through the Ministry of Education of the Republic of Korea and National Research

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Foundation of Korea (NRF) (2017078374). She is also grateful to the Korea Institute for Advanced Study (KIAS),

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where part of this work was done, during her sabbatical year in KIAS. The second author is partially supported

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by JSPS grants (17K05159, 17H02849, BBD30028). The third author was supported by the project I3346 of the

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Japan Society for the Promotion of Science (JSPS) and the Austrian Science Fund (FWF).

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