ap-5-10.dvi


Acta Polytechnica Vol. 50 No. 5/2010

Tilings Generated by Ito-Sadahiro and Balanced
(−β)-numeration Systems

P. Ambrož

Abstract

Let β > 1 be a cubic Pisot unit. We study forms of Thurston tilings arising from the classical β-numeration system and
from the (−β)-numeration system for both the Ito-Sadahiro and balanced definition of the (−β)-transformation.

Keywords: beta-expansion, negative base, tiling.

1 Introduction
Representations of real numbers in a positional numeration system with an arbitrary base β > 1, so-called
β-expansions, were introduced by Rényi [10]. During the fifty years since the publication of this seminal paper,
β-expansions have been extensively studied from various points of view.
This paper considers tilings generated by β-expansions in the casewhen β is a Pisot unit. A generalmethod

for constructing the tiling of a Euclidean space by a Pisot unit was proposed by Thurston [11], although an
example of such a tiling had already appeared in the work of Rauzy [9]. Fundamental properties of these tilings
were later studied by Praggastis [8] and Akiyama [1, 2].
In 2009, Ito and Sadahiro introduced a new numeration system [6], using a non-integer negative base

−β < −1. Their approach is very similar to the approach by Rényi. Another definition of a system using
a non-integer negative base −β < −1, obtained as a slight modification of the system by Ito and Sadahiro, was
considered by Dombek [4].
The main subject of this paper is to transfer the construction by Thurston into the framework of (−β)-

numeration (both cases) and to provide examples of how tilings (for fixed β) in the positive and negative case
can resembleand/ordiffer fromeachother. Thepaper is intendedasanentrypoint into a studyof theproperties
of these tilings.

2 Rényi β-expansions

Let β > 1 be a real number and let the transformation Tβ : [0,1) → [0,1) be defined by the prescription
Tβ(x) := βx − �βx�. The representation of a number x ∈ [0,1) of the form

x =
x1
β
+

x2
β2
+

x3
β3
+ · · · ,

where xi = �βT i−1β (x)�, is called the β-expansion of x. Since βT(x) ∈ [0, β) the coefficients xi (called digits)
are elements of the set {0,1, . . . , �β� − 1}.
The β-expansion of an arbitrary real number x ≥ 1 can be naturally defined in the following way: Find an

exponent k ∈ N such that
x

βk
∈ [0,1). Using the transformation Tβ derive the β-expansion of

x

βk
of the form

x

βk
=

x1
β
+

x2
β2
+

x3
β3
+ . . . ,

so that
x = x1β

k−1 + x2β
k−2 + . . . + xk−1β + xk +

xk+1
β
+ . . .

The β-expansion of x ∈ R+ is denoted by dβ(x), and as usual we write

dβ(x)= x1x2 . . . xk•xk+1xk+2 . . .

The digit string x1x2x3 · · · is said to be β-admissible if there exists a number x ∈ [0,1) so that dβ(x) =
•x1x2x3 . . . is its β-expansion. The set of admissible digit strings can be described using the Rényi expansion

7



Acta Polytechnica Vol. 50 No. 5/2010

of 1, denoted by dβ(1)= t1t2t3 . . ., where t1 = �β� and dβ(β − �β�)= t2t3t4 . . .. The Rényi expansion of 1 may
or may not be finite (i.e., ending in infinitely many 0’s which are omitted). The infinite Rényi expansion of 1,
denoted by d∗β(1) is defined by

d∗β(1)= lim
ε→0+

dβ(1 − ε) ,

where the limit is taken over the usual product topology on {0,1, . . . , �β� − 1}N. It can be shown that

d∗β(1)=

{
dβ(1) if dβ(1) is infinite,(
t1 · · · tm−1(tm − 1)

)ω
if dβ(1)= t1 · · · tm0ω with tm 
=0.

The characterization of admissible stings is given by the following theorem due to Parry.
Theorem 1 ([7]) A string x1x2x3 . . . over the alphabet {0,1, . . . , �β� − 1} is β-admissible, if and only if for
all i =1,2,3, . . .,

0ω �lex xixi+1xi+2 . . . ≺lex d∗β(1) ,

where �lex is the lexicographical order.

Using β-admissible digit strings, one can define the set of non-negative β-integers, denoted Zβ,

Zβ := {akβk + . . . a1β + a0 | ak · · · a1a00ω is a β-admissible digit string} ,

and the set Fin(β) of those x ∈ R+ whose β-expansions have only finitely many non-zero coefficients to the
right from the fractional point

Fin(β) :=
⋃

n∈N

1
βn

Zβ .

The distances between consecutive β-integers are described in [11]. It is shown that they take values in the

set {Δi | i =0,1, . . .}, where Δi =
∞∑

j=1

ti+j
βj
and dβ(1)= t1t2 . . .. Moreover, the sequence coding the distances

in Zβ is known to be invariant under a substitution provided dβ(1) is eventually periodic [5]. The form of this
substitution also depends on dβ(1).
If we consider β an algebraic integer, then obviously Fin(β) ⊂ Z[β−1]+. The converse inclusion, which is

very important for the construction of the tiling and also for the arithmetical properties of the system, does not
hold in general. An algebraic integer β for which

Fin(β)= Z[β−1]+

holds, is said to have Property (F).

3 Ito-Sadahiro (−β)-expansions

Now consider the real base −β < −1 and the transformation T−β :
[

−β
β +1

,
1

β +1

)
→
[

−β
β +1

,
1

β +1

)
defined

by the prescription

T−β(x)= −βx −
⌊

− βx +
β

β +1

⌋
.

Every number x ∈
[

−β
β +1

,
1

β +1

)
can be represented in the form

x =
x1
−β
+

x2
(−β)2

+
x3
(−β)3

+ · · · , where xi =
⌊

− βT i−1−β (x)+
β

β +1

⌋
.

The representation of x in such a form is called the (−β)-expansion of x and is denoted

d−β(x)= •x1x2x3 . . .

8



Acta Polytechnica Vol. 50 No. 5/2010

Byanalogy to the case ofRényi β-expansions, we use for the (−β)-expansion of x ∈ R a suitable exponent l ∈ N
such that

x

(−β)l
∈
[

−β
β +1

,
1

β +1

)
. It is shown easily that the digits xi of a (−β)-expansion belong to the set

{0,1, . . . , �β�}.
In order to describe strings that arise as (−β)-expansions of some x ∈

[
−β

β +1
,
1

β +1

)
, so-called (−β)-

admissible digit strings, we will use the notation introduced in [6]. We denote lβ =
−β

β +1
and rβ =

1
β +1

the

left and right end-point of thedefinition interval Iβ of the transformation T−β, respectively. That is Iβ = [lβ , rβ).
We also denote

d−β(lβ)= d1d2d3 . . .

Theorem 2 ([6]) A string x1x2x3 · · · over the alphabet {0,1, . . . , �β�} is (−β)-admissible, if and only if for all
i =1,2,3, . . .,

d−β(lβ) �alt xixi+1xi+2 ≺alt d∗−β(rβ) ,

where d∗−β(rβ)= lim
ε→0+

d−β(rβ − ε) and �alt is the alternate order.

Recall that the alternate order is definedas follows: We say that x1x2x3 . . . ≺alt y1y2y3 . . ., if (−1)i(xi −yi) >
0 for the smallest index i satisfying xi 
= yi. The relation between d∗−β(rβ) and d−β(lβ) is described in the same
paper.

Theorem 3 ([6]) Let d−β(lβ)= d1d2d3 . . . If d−β(lβ) is purely periodic with odd period-length, i.e., d−β(lβ)=
(d1d2 · · · d2l+1)ω, then d∗−β(rβ)= (0d1d2 · · · d2l(d2l+1 − 1))

ω. Otherwise, d∗−β(rβ)= 0d−β(lβ).

Similarly to the Rényi case, one can define the set of (−β)-integers, denoted Z−β, using the admissible digit
strings.

Z−β := {ak(−β)k + · · · a1(−β)+ a0 | ak · · · a1a00ω is a (−β)-admissible digit string} .

The set of distances between consecutive (−β)-integers has been described only for a particular class of β, cf. [3].

4 Balanced (−β)-numeration system

The last numeration system used in this paper is a slight modification of (−β)-numeration defined by Ito and

Sadahiro. Let −β < −1 be the base and consider the transformation S−β :
[
−
1
2
,
1
2

)
→
[
−
1
2
,
1
2

)
given by

S−β = −βx −
⌊

−βx +
1
2

⌋
.

The balanced (−β)-expansion of a number x ∈
[
−
1
2
,
1
2

)
, denoted dB,−β(x)= •x1x2x3 . . ., is

x =
x1
−β
+

x2
(−β)2

+
x3
(−β)3

+ . . . , where xi =

⌊
−βSi−1−β (x)+

1
2

⌋
.

Also in this casewe use for the (−β)-expansion of x ∈ R a suitable exponent l ∈ N such that
x

(−β)l
∈
[
−
1
2
,
1
2

)
.

It is shown easily that the digits xi of a balanced (−β)-expansion belong to the set
{

−
⌊

β +1
2

⌋
, . . . ,

⌊
β +1
2

⌋}
.

Note that sometimes d is used instead of −d.
A digit string x1x2x3 . . . is called balanced (−β)-admissible if it arises as the balanced (−β)-expansion of

some x ∈
[
−
1
2
,
1
2

)
. The two following theorems by Dombek [4] prove that also in this case the admissible

strings are characterized by the balanced (−β)-expansions of the endpoints of the interval
[
−
1
2
,
1
2

)
.

9



Acta Polytechnica Vol. 50 No. 5/2010

Theorem 4 ([4]) A string x1x2x3 . . . over the alphabet
{

−
⌊

β +1
2

⌋
, . . . ,

⌊
β +1
2

⌋}
is balanced (−β)-admis-

sible if and only if for all i =1,2,3, . . .

dB,−β

(
−
1
2

)
�alt xixi+1xi+2 . . . ≺alt d∗B,−β

(
1
2

)
,

where d∗B,−β

(
1
2

)
= lim

ε→0+
dB,−β

(
1
2

− ε
)
.

Theorem 5 ([4]) Let dB,−β

(
−
1
2

)
= d1d2d3 . . . Then

d∗B,−β(
1
2
)=

⎧⎪⎨
⎪⎩
(
d1 . . . d2l (d2l+1 − 1)d1 . . . d2l(d2l+1 − 1)

)ω
if dB,−β

(
−
1
2

)
=(d1 . . . d2l+1)

ω ,

d1 d2 d3 . . . otherwise.

The set of balanced (−β)-integers, denoted ZB,−β, is defined by analogy to the two previous cases.

ZB,−β := {ak(−β)k + . . . a1(−β)+ a0 | ak . . . a1a00ω is a balanced (−β)-admissible string} .

5 Constructing of the tiling

Recall that a Pisot number is an algebraic integer such that all its algebraic conjugates are in modulus strictly
smaller than one.
Let β > 1 be a Pisot number of degree d = r +2s. We denote β = β(1) and we assume that β(2), . . . , β(r)

are real conjugates of β and β(r+1), . . . , β(r+2s) are complex conjugates of β such that β(r+j) = β(r+s+j) for
j =1, . . . , s. Denote by x(j), j =1, . . . , n the corresponding conjugate of x ∈ Q(β), i.e.,

x = q0 + q1β + . . . + qd−1β
q−1 �→ x(j) = q0 + q1β(j) + . . . + qd−1(β(j))q−1 .

Consider the map Φ : Q(β) → Rd−1 defined by

Φ(x) :=
(
x(2), . . . , x(r), �(x(r+1)), �(x(r+1)), . . . , �(x(r+s)), �(x(r+s)) .

Proposition 6 ([1]) Let β > 1 be a Pisot number of degree d. Then Φ(Z[β]) is dense in Rd−1.

The map Φ is used to construct the tiling in the following way. Let w = w1 . . . wl ∈ {0,1, . . . , �β� − 1}∗ be
a finite word such that w0ω is an admissible digit string. We define the tile Tw as

Tw := {Φ(x) | x ∈ Fin(β) and (x)β = ak . . . x1x0•w1 · · · wl} .

The properties of the tiling of the Euclidean space using tiles Tw were described byAkiyama; the results are
summarized in the following theorems.

Theorem 7 ([1]) Let β be a Pisot unit of degree d with Property (F). Then

• Rd−1 =
⋃

w0ω admissible

Tw,

• for each x ∈ Zβ we have Φ(x) ∈ Inn(T�), where � is the empty word and Inn(X) denotes the set of inner
points of X; especially, the origin 0 is an inner point of the so-called central tile T�,

• for each tile Tw we have Inn(Tw)= Tw,

• ∂(Tw) is closed and nowhere dense in Rd−1, where ∂(Tw) is the set of boundary elements of Tw,

• if dβ(1)= t1 · · · tm−11 then each tile Tw is arc-wise connected.

10



Acta Polytechnica Vol. 50 No. 5/2010

Theorem 8 ([2]) Let β be a Pisot unit of degree d such that dβ(1) = t1 . . . tm(tm+1 · · · tm+p)ω with m, p the
smallest possible. Then there are exactly m + p different tiles up to translation.

Note that Q(β) = Q(−β) and Z[β] = Z[−β]. Thus the construction of the tiling associated to (−β)-
numeration follows the same lines, the corresponding mapping Φ− being defined using isomorphisms of the
extension fields Q(−β) and Q(−β(j)), and the following variant of Proposition 6 holds; its proof follows the
same lines as in the proof of the original proposition.

Proposition 9 Let β > 1 be a Pisot number of degree d. Then Φ−(Z[−β]) is dense in the space Rd−1.

6 Examples of tilings

In the rest of the paper we provide several examples of tilings associated with β cubic Pisot units, i.e., the
minimal polynomial of β is of the form x3 − ax2 − bx ± 1. Every time all the tiles Tw with w of length 0,1,2
are plotted.
So far no properties of tilings in the negative case similar to those in Theorem 7 and Theorem 8 have

been proved. However, the following examples demonstrate that it is reasonable to anticipate that most of
the properties remain valid. On the other hand, one can also observe that for a fixed β when we change the
β-numeration into the (−β)-numeration (either Ito-Sadahiro or balanced) the shape and form of the tiles can
be either preserved or changed slightly or completely.

6.1 Minimal polynomial x3 − x2 − 1

The tilings associated to −β are trivial in this case. Indeed, d−β(lβ)= 1001ω and dB,−β
(

−
1
2

)
=
(
10(−1)(−1)

(−1)(−1)(−1)010(−1)011
)ω
, hence Z−β = ZB,−β = {0} (cf. [3, 4]).

6.2 Minimal polynomial x3 − 2x2 − 2x − 1
This β is an example of a base for which the three considered tilings almost do not change. We have

dβ(1)=211 , d−β(lβ)=201
ω , dB,−β

(
−
1
2

)
=(1(−1)1)ω .

All three sets Zβ, Z−β and ZB,−β have the same set of three possible distances between consecutive elements,
namely {1, β − 2, β2 − 2β − 2}. The codings of the distances in these sets are generated by substitutions which
are pairwise conjugated. Recall that substitutions ϕ and ψ over an alphabet A are said to be conjugated if
there exists a word w ∈ A∗ such that ϕ(a) = wψ(a)w−1 for all a ∈ A. The tilings are composed of the same
tiles (up to rotation). See Figure 1.

6.3 Minimal polynomial x3 − 3x2 + x − 1
In this case

dβ(1)=2201 , d−β(lβ)= (201)
ω , dB,−β

(
−
1
2

)
=(1(−1)00)ω ,

and again all three sets of integers have the same possible distances between consecutive elements, Δi ∈
{1, β − 2, β2 − 2β − 2, β2 − 3β +1}. However, in this case the associated substitutions are not conjugated (the
condition is not fulfilled on exactly one of four letters) and even though the tilings do look similar, they are
composed of different tiles. See Figure 2.

6.4 Minimal polynomial x3 − 2x2 − 1
This β is an example of a base for which two tilings (and the corresponding properties of the sets of integers)
are very similar, but the third tiling differs substantially. We have

dβ(1)= 201 , d−β(lβ)= (2101)
ω , dB,−β

(
−
1
2

)
=(101)ω .

11



Acta Polytechnica Vol. 50 No. 5/2010

Rényi case Ito-Sadahiro case

Balanced case

Fig. 1: Minimal polynomial x3 − 2x2 − 2x −1

12



Acta Polytechnica Vol. 50 No. 5/2010

Rényi case

Ito-Sadahiro case

Balanced case

Fig. 2: Minimal polynomial x3 −3x2 + x − 1

13



Acta Polytechnica Vol. 50 No. 5/2010

Rényi case

Ito-Sadahiro case

Balanced case

Fig. 3: Minimal polynomial x3 − 2x2 −1

14



Acta Polytechnica Vol. 50 No. 5/2010

Rényi case Ito-Sadahiro case

Balanced case

Fig. 4: Minimal polynomial x3 − 3x2 +2x −1

The sets Zβ and ZB,−β have the same set of distances {1, β − 2, β2 −2β}, however the associated substitutions
are not conjugated. On the other hand there are five distances between consecutive elements in the set Z−β,
namely {1, β2 − β − 1, β − 1, β, β2 − β}. The forms of the tilings comply: the tiling in the Rényi case and the
tiling in the balanced case are somewhat similar, but the tiling in the Ito-Sadahiro case is completely different.
See Figure 3.

6.5 Minimal polynomial x3 − 3x2 +2x − 1
The last example demonstrates that the tiling can change fundamentally when considering different numeration
systems with fixed β. In this case

dβ(1)=201
ω , d−β(lβ)= (211)

ω , dB,−β

(
−
1
2

)
=(1010(−1)(−1)(−1)(−1)0(−1)0111)ω ,

there are three distances between consecutive elements in the set Zβ, four in the set Z−β and seven in the set
ZB,−β. The tilings are completely different. See Figure 4.

15



Acta Polytechnica Vol. 50 No. 5/2010

7 Conclusion
Due to the similar nature of β-numeration and (−β)-numeration, the transfer of the construction of the tiling
of a space due to Thurston into the framework of (−β)-numeration is quite straightforward.
In this paper we have provided several examples of these tilings (for both the Ito-Sadahiro definition and

the balanced definition of the −(β)-transformation). Although the shape and form of tiling can change dramat-
ically when one changes (for a fixed β) the β-numeration into the −(β)-numeration, in general the examples
demonstrate that the validity of most of the properties derived by Akiyama andPraggastis in the positive case
should be preserved. It remains an open question to provide proofs of such properties.

Acknowledgement

We acknowledge financial support from Czech Science Foundation grant 201/09/0584 and from grants MSM
6840770039 and LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic.

References

[1] Akiyama, S: Self affine tiling and Pisot numeration system. In Number theory and its applications (Kyoto,
1997), K. Győry and S. Kanemitsu, (eds.), vol. 2 of Dev. Math., Kluwer Acad. Publ. 1999, 7–17.

[2] Akiyama, S.: On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Japan 54,
2002, 283–308.

[3] Ambrož,P., Dombek,D.,Masáková,Z., Pelantová,E.: Numberswith integer expansions in the numeration
system with negative base. Submitted to Acta Arithmetica.

[4] Dombek, D.: Beta-numeration systems with negative base. Master’s thesis, Czech Technical University in
Prague, 2010.

[5] Fabre, S.: Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137, 1995, 219–236.

[6] Ito, S., Sadahiro, T.: Beta-expansions with negative bases. Integers, 9, 2009, A22, 239–259.

[7] Parry, W.: On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 1960, 401–416.

[8] Praggastis, B.: Markov partitions for hyperbolic toral automorphisms. PhD thesis, University of Washing-
ton, 1994.

[9] Rauzy, G.: Nombres algébriques et substitutions. Bull. Soc. Math. France, 110, 1982, 147–178.

[10] Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar,
8, 1957, 477–493.

[11] Thurston, W. P.: Groups, tilings, and finite state automata. AMS Colloquium Lecture Notes, 1989.

Ing. Petr Ambrož, Ph.D.
E-mail: petr.ambroz@fjfi.cvut.cz
Department of Mathematics FNSPE,
Czech Technical University in Prague
Trojanova 13, 120 00 Praha 2, Czech Republic

16