Fractal properties of number systems

Periodica Mathematica Hungarica Vol. 42 (1–2), (2001), pp. 51–68

FRACTAL PROPERTIES OF NUMBER SYSTEMS

Wolfgang Muller (Graz), Jorg M. Thuswaldner (Leoben)and Robert F. Tichy (Graz)

Dedicated to Professor Andras Sarkozy on the occasion of his 60th birthday

Abstract

In this paper we study properties of the fundamental domain F of numbersystems in the n-dimensional real vector space. In particular we investigate the frac-tal structure of its boundary ∂F . In a first step we give upper and lower bounds forits box counting dimension. Under certain circumstances these bounds are identicaland we get an exact value for the box counting dimension. Under additional as-sumptions we prove that the Hausdorff dimension of ∂F is equal to its box countingdimension. Moreover, we show that the Hausdorff measure is positive and finite.This is done by applying the theory of graph-directed self similar sets due to Fal-coner and Bandt. Finally, we discuss the connection to canonical number systemsin number fields, and give some numerical examples.

1. Introduction

Since Knuth [17, p. 608] investigated the so called “twin–dragon”, a planarset with fractal boundary which is associated with a certain number system in thering of Gaussian integers, several authors studied fractal sets related to numbersystems (cf. for instance [8, 7, 11, 12, 20, 21, 23]). All these sets are in some wayrelated to fundamental domains of number systems. The aim of this paper is asystematic description of the geometric properties of number systems in Zn, as wellas an extension and refinement of the known results in this field. In particular, wecalculate the Hausdorff dimension of the boundary of the corresponding fundamentaldomains, and we estimate the related Hausdorff measure.

Let B ∈ Zn×n be a matrix whose eigenvalues have all modulus greater than1. Furthermore, let N ⊆ Zn be a complete residue system modB which contains0. The pair (B,N ) is called a number system in Zn if every m ∈ Zn admits arepresentation of the form

m =k∑j=0

Bjaj (aj ∈ N ).

Mathematics subject classification numbers: 11A63, 28A78, 28A80.Key words and phrases: number systems, Hausdorff dimension, fractals.

0031-5303/01/$5.00 Akademiai Kiado, Budapestc© Akademiai Kiado, Budapest Kluwer Academic Publishers, Dordrecht

52 w. muller, j. m. thuswaldner and r. f. tichy

Since N is a complete residue system modB this representation is unique. Thenumber of digits aj ∈ N in such a number system is |N | = [Zn : BZn] = |detB| > 1.The set

F :={∑j≥1

B−jaj | aj ∈ N}⊆ Rn

is called the fundamental domain of (B,N ). We are mainly interested in the fractalproperties of the boundary ∂F of F . For the investigation of these properties itis not neccessary to assume that (B,N ) is a number system. The analysis of thispaper applies to the wider class of pairs (B,N ) which are just touching coveringsystems. Katai [12] calls a pair (B,N ) a just touching covering system (JTCS forshort) if

λ((m1 + F) ∩ (m2 + F)) = 0 (m1 6= m2, m1,m2 ∈ Zn) .

Here λ denotes the Lebesgue measure in Rn.In Section 2 we collect basic facts about JTCSs. One important feature of

a JTCS is that the boundary of its fundamental domain can be described by agraph G(S). This graph is constructed in Section 3. In Section 4 we give lowerand upper bounds of the box counting dimension of ∂F . These lower and upperbounds coincide if all eigenvalues of B have the same modulus. Under additionalassumptions we prove in Section 5 that the box counting dimension is equal to theHausdorff dimension. Moreover, we show that the Hausdorff measure is positiveand finite. To do this we apply the theory of graph-directed self similar sets in thesense of Falconer [6, Chapter 3] and Bandt [1].

Finally, in Section 6, we give some numerical examples of our results anddiscuss the connection to canonical number systems in number fields, which are thenatural extensions of radix expansions to algebraic numbers.

For the convenience of the reader we recall the different notions of dimensionwhich are used in this paper. For a non-empty bounded subset E of Rn let Mr(E)be the smallest number of closed balls of radius r that cover E. The lower andupper box counting dimensions of E are defined by

dimB(E) = lim infr→0

logMr(E)− log r

and dimB(E) = lim supr→0

logMr(E)− log r

,

respectively. If the limit exists, the common value of the lower and upper boxcounting dimensions is called box counting dimension. The value of these dimensionsis not altered if one replaces Mr(E) by Mr(E), the smallest number of sets ofdiameter r that cover E (cf. Falconer [6, p. 20]). We will use both variants in thispaper. The Hausdorff dimension of a Borel set E is defined as follows: Let {Ui}∞i=1,be a δ-cover of E, i.e. E ⊆

⋃∞i=1 Ui and diam(Ui) < δ where diam(Ui) denotes the

diameter of Ui. Then the s-dimensional Hausdorff measure of E is given by

Hs(E) = limδ→0

(inf{ ∞∑i=1

diam(Ui)s : {Ui}∞i=1 is a δ-cover of E}).

fractal properties of number systems 53

The Hausdorff dimension of E is now defined by

dimH(E) = inf{s : Hs(E) = 0} = sup{s : Hs(E) =∞}.

These different notions of dimension are related by the following inequalities:

dimH(E) ≤ dimB(E) ≤ dimB(E).(1)

For a further discussion of these and other kinds of dimensions we refer to Fal-coner [6].

2. Just touching covering systems

Let B ∈ Zn×n be a matrix whose eigenvalues have all modulus greater than1, and let N ⊆ Zn be a complete residue system modB which contains 0. To decidewhether or not (B,N ) is a number system we introduce the map

J : Zn −→ Znx 7−→ B−1(x− a) .

Here a denotes the unique element of N such that x− a ∈ BZn. Furthermore, wechoose any norm in Rn such that ||B−1|| < 1. A possible choice is the Euclideannorm for which ||B−1|| = max1≤i≤n |λi|−1, where λi denotes the eigenvalues of B.Set

K := maxa∈N||a||, L := K(||B−1||−1 − 1)−1 .(2)

It follows from ||J(x)|| ≤ ||B−1||(||x|| +K) that

||J(x)|| < ||x|| if ||x|| > L ,||J(x)|| ≤ L if ||x|| ≤ L .

This proves that the set P := {m ∈ Zn | ∃k ∈ N : Jk(m) = m} of periodic elementsis finite. Denote by l = l(m) ≥ 0 the least integer such that J l(m) = p ∈ P . Wefind that every m ∈ Zn has a unique representation of the form

m =l−1∑j=0

Bjaj +Blp (aj ∈ N , p ∈ P)(3)

with J l−1(m) = al−1 + Bp 6∈ P if l ≥ 1. (B,N ) is a number system if and only ifP = {0}. This gives an algorithmic method to decide whether or not a given pair(B,N ) is a number system.

To study the properties of the fundamental domain

F ={∑j≥1

B−jaj | aj ∈ N}⊆ Rn


and of its boundary ∂F , it is not neccessary to assume that (B,N ) is a numbersystem. Denote by

D :={ k∑j=0

Bjaj | k ≥ 0, aj ∈ N}

(4)

the set of all properly representable elements of Zn.

Proposition 2.1. The fundamental domain of (B,N ) has the following prop-erties

(i) F is compact.(ii) F has interior points. More specifically

⋃p∈P(p+F) contains a neighbourhood

of the origin.(iii) Rn =

⋃m∈Zn(m+ F) .

(iv) λ((m1 + F) ∩ (m2 + F)) = 0 if m1 6= m2 and m1,m2 ∈ D.(v) Let Ek = {B−k(m + int(F)) | m ∈ Zn} and E =

⋃k≥0Ek. Then E forms a

net, i.e. for F1, F2 ∈ E either F1 ⊆ F2 or F2 ⊆ F1 or F1 ∩ F2 = ∅.

Proof. For the case of number systems in Z a proof is given in Katai [12]; seealso [15]. Similar problems are treated in a much more general context in Lagarias–Wang [19] and Vince [24]. In our case the arguments run along the same lines, andthus we give only some comments. Assertion (i) can be shown by applying Cantor’sdiagonal principle. Assertion (ii) is a special case of Vince [24, Theorem 1] (cf. also[19, Theorem 1.1]). Assertions (iii) and (iv) are easy consequences of assertion (ii).Assertion (v) follows from the definition of F . �

By property (iv) every number system is a JTCS. Katai [12] proved that(B,N ) is a JTCS if and only if every element of Zn is the difference of two properlyrepresentable elements, i.e. Zn = D − D. The last condition is equivalent toP ⊆ D −D (cf. Theorem 3.6).

3. The graphs G(S) and G(S)

For a JTCS the boundary ∂F has a description by means of a graph. Set

S := {m ∈ Zn \ {0} |F ∩ (m+ F) 6= ∅} , S0 := S ∪ {0} ,

Bm := F ∩ (m+ F) , ∆F :=⋃m∈S

Bm .

Lemma 3.1. If (B,N ) is a JTCS, then ∂F = ∆F .

Proof. Since (B,N ) is a JTCS, we have λ(Bm) = 0 for m 6= 0. Thus Bmcontains no interior points of F . This proves ∆F ⊆ ∂F . On the other hand

∂F = F ∩ Fc ⊆ F ∩ (∑m6=0

(m+ F)) = ∆F .

This implies ∂F = ∆F . �


In the following we construct a graph G(S) which describes ∆F .

Lemma 3.2. Set B = N −N . Then(i) m ∈ S0 if and only if m has the representation

m =∞∑j=1

B−jbj with bj ∈ B.

(ii) x ∈ Bm if and only if m is representable as in (i) and

x =∞∑j=1

B−jaj with aj − a′j = bj ; aj, a′j ∈ N .

Proof. Every x ∈ Bm = F ∩ (m+ F) admits two representations

x =∞∑j=1

B−jaj and x−m =∞∑j=1

B−ja′j with aj , a′j ∈ N .

Thus m has the representation

m =∞∑j=1

B−j(aj − a′j) =∞∑j=1

B−jbj

with bj ∈ B. This implies (i) and (ii). �Let G(Zn) be a labeled directed graph with set of vertices Zn and set of labels

B. The labeled edge connecting two vertices m1 and m2 is defined as follows:

m1b−→ m2 :⇐⇒ Bm1 −m2 = b ∈ B (m1,m2 ∈ Zn).(5)

Let G(S) and G(S0) be the restrictions of G(Zn) to S and S0, respectively.

Proposition 3.3. The graph G(Zn) has the following properties:

(i) m1b−→ 0 =⇒ m1 = 0.

(ii) m2 ∈ S0 and m1b−→ m2 =⇒ m1 ∈ S0.

(iii) m2 ∈ S0 =⇒ ∃ m1 ∈ S0, b ∈ B such that m1b−→ m2.

(iv) m1 ∈ S0 =⇒ ∃ m2 ∈ S0, b ∈ B such that m1b−→ m2.

(v) m1b1−→ m2

b2−→ · · · br−1−−−→ mrbr−→ m1 =⇒ m1 ∈ S0.

(vi) m ∈ S0 =⇒ ‖m‖ ≤ 2L, where L is defined in (2).

Proof.

(i) m1b−→ 0 means Bm1 = b ∈ B. Thus, b ≡ 0(modB). This implies b = 0 and

m1 = 0.(ii) By Lemma 3.2 (i) every m2 ∈ S0 admits a representation m2 =

∑j≥1 B

−jbj

with bj ∈ B. Now m1b−→ m2 implies m1 = B−1b +

∑j≥2 B

−jbj−1. Againusing Lemma 3.2 (i), yields m1 ∈ S0.


(iii) For m2 ∈ S0 choose b ∈ N such that m2 + b ≡ 0(modB) and set m1 =

B−1(m2 + b) ∈ Zn. Then m1b−→ m2. Applying Lemma 3.2 (i) to m2 and m1

yields m1 ∈ S0.(iv) Every m1 ∈ S0 admits a representation

∑j≥1B

−jbj with bj ∈ B. Set b = b1

and m2 = −b+Bm1. By Lemma 3.2 (i) m2 ∈ S0 and m1b−→m2.

(v) m1b1−→ m2

b2−→ . . .br−1−−−→ mr

br−→ m1 implies

m1 =r∑i=1

B−ibi +B−rm1 =∑l≥0

B−rl( r∑i=1

B−ibi)∈ S0.

(vi) For m ∈ S0 and K as in (2) we have

‖m‖ =∥∥∥∑j≥1

B−jbj

∥∥∥ ≤∑j≥1

‖B−1‖j2K ≤ 2L. �

Remark 3.4. The above proposition provides an algorithm for constructingthe graphG(S0). G(S0) is the union of all cycles inG(Zn) and of all walks connectingtwo of these cycles. By Proposition 3.3 (vi) S0 is contained in the finite set {m ∈Zn | ‖m‖ ≤ 2L}.

Any infinite walk m b1−→ m2b2−→ m3

b3−→ . . . in G(S0) yields a representation

m =∞∑j=1

B−jbj .

On the other hand each such representation yields an infinite walk inG(S0), startingat m. Together with Lemma 3.2 (ii) this allows one to determine all points of Bm.To do this in an algorithmic way, we construct a new graph G(S0). For b ∈ B set

A(b) := {a ∈ N | ∃a′ ∈ N with a− a′ = b}and π(b) = |A(b)|. The graph G(S0) results from G(S0) by substituting each edge

m1b−→ m2 by π(b) edges m1

a−→ m2, with a ∈ A(b). Then each walk ma1−→ m2

a2−→. . . in G(S0) starting at m yields a point

x =∞∑j=1

B−jaj ∈ Bm,

and each point x ∈ Bm can be constructed in this way.

Lemma 3.5. For each m ∈ S0 there are exactly |N | edges in G(S0) leading tom. They have pairwise different labels. This implies that every label a ∈ N occursexactly once.

Proof. We have to prove that for every m ∈ S0 and a ∈ N there existsexactly one m1 ∈ S0 with Bm1 −m = a− a′, where a′ ∈ N . For given m ∈ S0 and


a ∈ N select a′ ∈ N such that a′ ≡ m+a(modB). Then m1 = B−1(m+a−a′) ∈ Zn

and m1a−a′−−−→ m in G(S0). Thus m1 ∈ S0 by Lemma 3.2 (i). This proves the

existence of such an m1. The uniqueness follows from the fact that a′ ∈ N isuniquely defined by a′ ≡ m+ a(modB). �

The graph G(S0) admits an easy criterion to check whether (B,N ) is a JTCSor not.

Theorem 3.6. The following assertions are equivalent:(i) (B,N ) is a JTCS.(ii) For each m ∈ S there exists a walk in G(S0) leading from 0 to m.(iii) Zn = D −D, where D is defined as in (4).

Proof. The equivalence of (i) and (iii) is proved in Katai [12]. The equiva-lence with (ii) is implicitly contained in Katai’s proof. �

In the following sections we need the graph G(S) which is the restriction ofG(S0) to the set of vertices S.

4. The box counting dimension of ∂F

In the remaining part of this paper we assume that the graph G(S) has anaccompanying matrix P with a unique (positive real) eigenvalue of largest modulus.By the Perron–Frobenius Theorem (cf. Seneta [22, p. 1]) this is always the case, ifG(S) is a primitive graph. A finite directed graph is called primitive, if it is stronglyconnected and the greatest common divisor of the lengths of its closed directed walksis 1 (cf. Brualdi–Ryser [3, p. 68]).

Now we start with the computation of the box counting dimension of theboundary of F . Let Wk(m) be the set of walks of length k in G(S) starting atm ∈ S and let

Wk :=⋃m∈S

Wk(m) .(6)

Then

|Wk| =∑m∈S|Wk(m)|(7)

is the total number of walks of length k in the graph G(S). By the definition ofπ(b) we have

|Wk+1(m1)| =∑m2∈S

m1b−→m2

π(b)|Wk(m2)|

and (|W0(m)|)m∈S = (1, . . . , 1)t. We obtain the recurrence

(|Wk+1(m)|)m∈S = P (|Wk(m)|)m∈S


with the accompanying matrix P ∈ N|S|×|S|0 of G(S) defined by

Pm1,m2 =

{π(b) m1

b−→m2

0 otherwise(m1,m2 ∈ S).(8)

Let µmax be the (real and positive) eigenvalue of largest modulus of P . Then thereis a constant c > 0 such that

|Wk| =∑m∈S|Wk(m)| = cµkmax + o(µkmax).(9)

Proposition 4.1. Let (B,N ) be a JTCS and set

Qk = {x ∈ B−kZn | (x+B−kF) ∩ ∂F 6= ∅}.(10)

Then

C1|Wk| ≤ |Qk| ≤ C2|Wk|,

where C1 and C2 do not depend on k.

Proof. First we prove |Wk| ≤ |S||Qk|. To do this, we consider the mappingwhich sends every walk m1

a1−→ m2a2−→ . . .

ak−→ mk+1 in Wk to x =∑kj=1 B

−jaj ∈Qk. We have to prove that this mapping sends at most |S| walks to a fixed x ∈ Qk.By the uniqueness of the digits representation in (B,N ) every x ∈ Qk correspondsto exactly one labeling (a1, . . . , ak). By Lemma 3.5 there is at most one walk withlabeling (a1, . . . , ak) ending at a fixed m in |S|. Thus there are at most |S| differentwalks in G(S) having this labeling.

In order to prove |Qk| ≤ C2|Wk|, we define a mapping which sends every x ∈Qk to a walk in Wk. Let x ∈ Qk. Then there exists an element z ∈ x+B−kF ∩∂F .Since F is closed, it follows that z ∈ F . Hence, there exist aj ∈ N , j ≥ 1, with

z =∞∑j=1

B−jaj ∈ ∂F .

If there is more than one sequence (a1, a2, . . . ) that can be obtained by this con-struction we select that sequence, which is smallest in terms of the component-wiselexicographic order on the set of all sequences of labels. Since z ∈ ∂F , there existwalks in G(S) with labeling (a1, a2, . . . ). To fix a uniquely determined choice of thesewalks we select that one having smallest end state (in terms of any order on S). Thiswalk is considered to be the image of x. We have to prove that this mapping sendsat most C2 elements x ∈ Qk to the same walk m1

a1−→ m2a2−→ . . .

ak−→ mk+1 ∈ Wk.Define

zk =k∑j=1

B−jaj .

Then z = zk + B−kf1 and z = x + B−kf2 with f1, f2 ∈ F . Thus x ∈ zk + B−kE ,


where E = {f1 − f2 | f1, f2 ∈ F} is a compact set. By Proposition 2.1 (iii) it can becovered by a finite number of translates of F . Multiplication by B−k yields

B−kE ⊆ (c1 +B−kF) ∪ . . . ∪ (cl +B−kF),

for l ∈ N independent of k and c1, . . . , cl ∈ B−kZn. Select j ≤ l such that x ∈(zk + cj) +B−kF . Since obviously x ∈ x+B−kF we conclude that(

(zk + cj) +B−kF)∩(x+B−kF

)6= ∅.(11)

For fixed zk and cj there are at most |S| + 1 distinct points x ∈ Qk suchthat (11) holds. Since for j there are l possibilities, we find that there are at mostC2 = (|S|+1)l distinct points x ∈ Qk yielding the same walk m1

a1−→ m2a2−→ . . .

ak−→mk+1 ∈Wk. �

Lemma 4.2 (cf. Barnsley [2, p. 176, Theorem 1]). Let E be a bounded subsetof Rn. For real numbers 0 < r < 1 and c > 0 set εk = crk, k ≥ 1. Then

dimB(E) = limk→∞

( logMεk(E)− log εk

)= limk→∞

( log Mεk(E)− log εk

).

Analogous results hold for dimB(E) and dimB(E).

Theorem 4.3. Let (B,N ) be a JTCS such that G(S) has an accompanyingmatrix P with a unique (real positive) eigenvalue µmax of largest modulus. Denote byλmin and λmax the eigenvalues of minimal and maximal modulus of B, respectively.Then the lower and upper box counting dimensions of ∂F satisfy the inequalities

logµmax

log |λmax|≤ dimB∂F ≤ dimB∂F ≤

logµmax

log |λmin|.

Proof. Since (B,N ) is a JTCS, by Proposition 2.1 (i), a ball of radius 1intersects at most a finite number, say C, of sets of the form m + F for m ∈ Zn.Since λ−1

max is the eigenvalue of B−1 having smallest modulus we conclude that aball of radius r = |λmax|−k intersects at most C of the sets B−k(m+ F), m ∈ Zn.From the definitions of Qk and Mr(∂F) we obtain

C−1|Qk| ≤M|λmax|−k(∂F).(12)

By Lemma 4.2, Proposition 4.1 and (9)

dimB(∂F) ≥ lim infk→∞

log(C−1|Qk|)log(|λmax|k)

= lim infk→∞

log |Pk|log(|λmax|k)

=logµmax

log |λmax|.

Since F is compact and λ−1min is the eigenvalue of largest modulus of B−1, there

exists a constant f , such that diam(B−kF) ≤ f |λmin|−k. From the definitions ofQk and Mr(∂F)

Mf |λmin|−k(∂F) ≤ |Qk|.(13)


Hence, by Lemma 4.2, Proposition 4.1 and (9)

dimB(∂F) ≤ lim supk→∞

log(|Qk|)log(|λmin|k)

= lim supk→∞

log |Pk|log(|λmin|k)

=logµmax

log |λmin|. �

5. The Hausdorff dimension of ∂F

In order to determine the Hausdorff dimension of ∂F and to estimate theHausdorff measure of ∂F we apply the theory of graph-directed self similar sets inthe sense of Falconer [6] and Bandt [1]. This theory extends Hutchinson’s theory ofself similar sets, see for instance [4, 5, 10]. In this section we give a short accountof this theory. For details and proofs we refer to Chapter 3 of Falconer’s book.

Let d be a metric on Rn. A mapping ϕ : Rn −→ Rn is called a contraction iffor some constant c < 1

d(ϕ(x), ϕ(y)) ≤ cd(x, y) (x, y ∈ Rn).

We call the infimum of the constants c < 1, for which this inequality holds the ratioof the contraction ψ. A contraction, which maps any subset of Rn to a geometricallysimilar (homothetic) set is called a contracting similitude.

Let G be a primitive labeled directed graph with set of vertices {1, 2, . . . , q},set of edges E, and set of labels A. As usual, an edge from vertex i to vertex jlabeled by a is denoted by i

a−→ j. For each edge e ∈ E let ϕe be a contractingsimilitude of ratio re with 0 < re < 1. Then there is a unique family of non-empty,compact sets F1, . . . , Fq, such that

Fi =q⋃j=1

⋃e: i

a−→j

ϕe(Fj) (1 ≤ i ≤ q),

where the union is extended over all edges e : i a−→ j. {F1, . . . , Fq} is called a familyof graph-directed sets and F := F1 ∪ . . . ∪ Fq a graph-directed self similar set.

For α ∈ R define the matrix (rαi,j)i,j,=1,... ,q by ri,j =∑e: i

a−→jrαe . Then the

value of α, for that the Perron–Frobenius eigenvalue of the matrix (rαi,j) equals oneis called the similarity dimension of Fi (1 ≤ i ≤ q).

We say that the open set condition holds for the family of graph-directed setsF1, . . . , Fq, if there exist bounded non-empty open sets Vi, 1 ≤ i ≤ q, such that

q⋃j=1

⋃e: i

b−→j

ϕe(Vj) ⊆ Vi (1 ≤ i ≤ q),(14)

where these unions are disjoint for all i.Denote by Pk(m) the set of walks in G of length k ending at m. If P ∈ Pk(m)

has labeling (e1, . . . , ek), then write ϕP := ϕe1 ◦ . . . ◦ ϕek . Let δ > 0 and let Pbe a path in G. Denote by P ′ the walk that emerges from P by leaving away thefirst edge. Then we say that ϕP (Fj) has diameter almost δ, if diam(ϕP (Fj)) ≤


δ <diam(ϕP ′(Fj)). With these definitions we are able to state the following generalresults, which we use for the determination of the Hausdorff dimension of ∂F .

Theorem 5.1 (cf. [1, Theorem 9.2]). Let F1, . . . , Fq be a family of graph-directed sets with strongly connected graph G. Then the following assertions areequivalent:

(i) Let A be a set with diameter δ > 0 and j ∈ {1, . . . , q}. Let Cδ be the numberof walks P in G, for which ϕP (Fj) intersects A and has diameter almost δ.Then Cδ is uniformly bounded in δ.

(ii) The open set condition (14) holds for the family F1, . . . , Fq.

Theorem 5.2 (cf. [6, Corollary 3.5]). Let F1, . . . , Fq be a family of graph-directed sets which fulfills the open set condition (14). Then there is a number ssuch that dimB(Fi) = dimH(Fi) = s and 0 < Hs(Fi) <∞ for 1 ≤ i ≤ q.

Proposition 5.3. Let (B,N ) be a JTCS such that G = G(S) is primitive.Assume that the eigenvalues of B have all the same modulus λ > 1. Then themappings

ϕa(x) = B−1(x+ a) (a ∈ N ).(15)

are contracting similitudes with ratio λ−1. For all n ∈ S we have

Bn =⋃

a∈N ,m∈Sna−→m

ϕa(Bm).

Hence, ∂F is a graph-directed self similar set.

Proof. If x ∈ Bm and na−→ m in G(S) then ϕa(x) ∈ Bn by Lemma 3.2 (ii).

On the other hand every x ∈ Bn admits a representation

x =∑j≥1

B−jaj ,

where n a1−→ ma2−→ . . . is a walk in G(S). Hence,

x′ :=∑j≥1

B−jaj+1 ∈ Bm

satisfies ϕa1(x′) = x. �

Theorem 5.4. Let (B,N ) be a just touching covering system in Zn and as-sume that all eigenvalues of B have the same modulus λ. Assume further that theassociated graph G(S) is primitive and denote by µmax the unique (positive real)


eigenvalue of largest modulus. Then its Hausdorff dimension is equal to its boxdimension and is given by

s =log µmax

logλ.

The s-dimensional Hausdorff measure Hs(∂F) is positive and finite.

Remark 5.5. Since the s-dimensional Hausdorff measure of ∂F is positiveand finite, its Besicovitch dimension is equal to xs.

Proof. We check condition (i) of Theorem 5.1. A special feature of thecontracting similitudes ϕa is that they all have the same ratio of contraction λ−1.Thus if P is a walk in G(S) of length k then diam(ϕP (Bm)) = λ−kdiam(Bm). Ifδ ≥diam(Bm) there is no walk P for which ϕP (Bm) has diameter almost δ. If 0 <δ <diam(Bm) then ϕP (Bm) has diameter almost δ if and only if P has length k0 =k0(δ) = dlog(diam(Bm)/δ)/ logλe. For any A ⊆ Rn with diameter δ <diam(Bm)and any m′ ∈ S we have to prove that there is a constant C independent of A andδ such that ∣∣ {P ∈ Pk0(m′) |ϕP (Bm) ∩A 6= ∅}

∣∣ ≤ C.By Lemma 3.5 there exists at most one walk P in Pk0(m′) with labeling

(a0, . . . , ak0−1). It follows that∣∣ {P ∈ Pk0(m′) |ϕP (Bm) ∩A 6= ∅}∣∣

≤∣∣ {(a0, . . . , ak0−1) ∈ N k0 |ϕa0 ◦ . . . ◦ ϕak0−1(Bm) ∩A 6= ∅

} ∣∣≤

∣∣∣∣∣∣(a0, . . . , ak0−1) ∈ N k0

∣∣∣∣B−k0

Bm +k0−1∑j=0

Bjaj

∩A 6= ∅∣∣∣∣∣∣

≤∣∣∣ {l ∈ Zn |B−k0(Bm + l) ∩A 6= ∅

} ∣∣∣≤∣∣∣ {l ∈ Zn | F + l ∩Bk0A 6= ∅

} ∣∣∣.Since diam(Bk0A) ≤ δλk0 ≤ λdiam(Bm) ≤ λdiam(F) and since {F + l}l∈Zn tesse-lates Rn we conclude that there is a constant C depending only on F and λ suchthat ∣∣∣ {l ∈ Zn | F + l ∩Bk0A 6= ∅

} ∣∣∣ ≤ C.By Theorem 5.1 the family {Bm}m∈S satisfies the open set condition and Theo-rem 5.2 is applicable. Since all similitudes have ratio λ−1 the similarity dimensionof Bm is logµmax/ logλ = dimB(Bm). This completes the proof. �


6. Applications

In this section we want to give a few examples of just touching coveringsystems. An important class of examples is given by canonical number systems inalgebraic number fields.

Definition 6.1. Let K be a number field and ZK its ring of integers. Letb ∈ ZK and N = {0, 1, . . . , |N(b)| − 1}, where N(b) denotes the norm of b over Q.The pair (b,N ) is called a canonical number system in K, if each γ ∈ ZK admits arepresentation of the form

γ = c0 + c1b+ c2b2 + · · ·+ chb

h(16)

with h ∈ N0 and ci ∈ N (0 ≤ i ≤ h). b is called the base of the canonical numbersystem (b,N ).

Remark 6.2. One obtains a little bit more general notion of number systemsin number fields if one replaces N by an arbitrary complete residue system modulob (cf. for instance Grochenig–Haas [8]).

Canonical number systems were introduced by Katai and Szabo [16] and lateron studied by various authors ([13, 14]). Using a result due to Gyory [9], Kovacsand Petho [18] established an algorithm which decides whether or not there existsa canonical number system in a given number field K and computes all possiblebases. Furthermore, they showed that for each base b all the (algebraic) conjugatesσj(b) have modulus > 1.

A canonical number system (b,N ) in a number field K of order n induces anumber system in Zn in the following way. Consider the embedding Φ : K −→ Rnwhich maps

∑n−1j=0 αjb

j ∈ K to (α0, . . . αn−1) ∈ Rn. (16) implies Φ(ZK) = Zn. Letmb(x) = xn + dn−1x

n−1 + · · · + d1x + d0 be the minimal polynomial of b over Q.Then define the matrix

B =

0 0 · · · 0 0 −d01 0 · · · 0 0 −d10 1 · · · 0 0 −d2· · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · ·0 0 · · · 0 1 −dn−1

.

Since Φ(bx) = BΦ(x) the matrix B is the base of a number system with digit setN := {ce1 | 0 ≤ c < N} where e1 = (1, 0, . . . , 0)t ∈ Rn and N = d0 = |detB| =|N(b)|. Since the eigenvalues of B are the conjugates of b we conclude that alleigenvalues of B have modulus greater than one.

It is a fairly easy observation that the fields Q(d1/n), d ∈ Z, are the onlyalgebraic number fields of degree n with an integral basis {1, b, . . . , bn−1} such thatall conjugates σj(b) have equal moduli. In the case of cyclotomic fields (i.e. |d| = 1)we have |σj(b)| = 1, which is excluded in the assumptions of Theorem 5.4.


In the case of a quadratic number field Q(√d), d squarefree, it is well known

that the possible choices of b such that {1, b} forms an integral basis are b = α+√d,

α ∈ Z for d 6≡ 1(mod 4) and b = α+√d

2 , α ≡ 1(mod 2) for d ≡ 1(mod 4). Theconjugates of b have the same modulus if and only ifQ(

√d) is an imaginary quadratic

number field. In the remaining number fields Q(d1/n), |d| > 1, the only possiblechoices are of the form b = ζd1/n with an n-th root of unity ζ. In the case b = d1/n

a simple computation shows that the corresponding fundamental domain is an n-dimensional cube.

Since we are interested in fractal properties of fundamental domains we willrestrict ourselves to the imaginary quadratic case. As a typical example we considercanonical number systems in the Gaussian number field Q(i).

In [16] all possible bases of canonical number systems in Q(i) were shown to beof the form b = −a± i with a ∈ N. In this case the minimal polynomial of the baseb is mb(x) = x2 + 2ax+ a2 + 1 and the set of digits is given by N = {0, 1, . . . , a2}.The corresponding number system in Z2 is defined by the matrix

B =(

0 −1− a2

1 −2a

).

We now construct the graph G(S). According to the definition of the graph G(Z2)we have (

m1m2

)βe1−−→

(n1n2

)⇐⇒

{n2 = m1 − 2am2

β = −n1 − (a2 + 1)m2(|β| ≤ a2).(17)

Since each element of S has a successor, the elements of S have the form

[k, l] :=(k + 2al

l

)(k, l ∈ Z).

In this notation (17) reads

[k1, l1]βe1−−→ [k2, l2]⇐⇒

{k1 = l2,β = −k2 − 2al2 − (a2 + 1)l1.

Note that [k, l] ∈ S0 satisfies |k| < c if and only if |l| < c. This follows from the factthat each vertex of G(S0) has a successor and a predecessor. From (2) we obtain

K = a2, ‖B−1‖ =1|b| = (a2 + 1)

12 (λ1 = b, λ2 = b)

L = K‖B−1‖

1− ‖B−1‖ =K

|b| − 1=

a2√a2 + 1− 1

< a+32.

By Proposition 3.3 (vi) an element [k, l] ∈ S0 satisfies ||[k, l]||2 ≤ 2L < 2a+ 3. Thisimplies |k|, |l| < 2a+ 3 and |k + 2al| < 2a+ 3. Hence, |l| < 2 + 3

a , and this implies|l|, |k| ≤ 2 for a ≥ 3. Using |k + 2al| < 2a + 3 with |k| ≤ 2 we obtain the sharperbound |l|, |k| ≤ 1 for a ≥ 3. In the case a = 2 the above considerations yield thebounds |k|, |l| ≤ 2. In the case a = 1 one obtains |k|, |l| ≤ 4.


[0,-1] [-1,0]

[1,0] [0,1]

[-1,1] [1,-1]

1-2a (a-1)2

-(a-1)2 2a-1

(a-1) -12

-(a-1) +12

a2

-2a

-a2

2a

Fig. 1. The graph G(S)

Following Section 3 it is now easy to construct the graph G(S). For a ≥ 3 ithas the form indicated in Figure 1.

In the case a = 1 a lengthy, but easy direct calculation using the bounds|k|, |l| ≤ 4 shows that G(S) results from the graph of Figure 1 by cancellation ofthe two edges labeled with ±2a. In the exceptional case a = 2 a direct computationshows that G(S) is the graph drawn in Figure 2.

-4 4

4 -4

31

-3-1

2

-2

2 -2

-3 3

1-1-4

4

[1,0] [0,-1] [0,1] [-1,0]

[-1,1] [1,-1]

[2,-1] [-2,1]

[-2,2] [2,-2]

Fig. 2. The graph G(S)

In order to establish an explicit formula for the Hausdorff dimension of ∂Fwe need the eigenvalue µmax. For a 6= 2 the characteristic polynomial of P is given


by

cP (x) = (x− 1)(x2 + 2ax+ a2 + 1)(x3 − (2a− 1)x2 − (a− 1)2x− (a2 + 1)).

The eigenvalue µmax is the dominating root of the cubic factor. Since λ = 1|b| =

2√a2+1

, Theorem 5.4 yields in the case a 6= 2

dimH(∂F) = dimB(∂F) = 2logµmax

log(a2 + 1)=: Da.(18)

Furthermore, Theorem 5.4 implies that the corresponding Hausdorff measure ispositive and finite.

In the exceptional case a = 2 Theorem 5.4 is not applicable since the graphdrawn in Figure 2 is not primitive. A direct computation shows that the maximaleigenvalue µmax of the accompanying matrix G(S) is again the dominating root ofcP (x) as in the case a 6= 2. By Theorem 4.3 the box dimension of ∂F is equal to D2.We further observe that the graph of Figure 1 is a primitive subgraph of the graphdrawn in Figure 2. By Theorem 5.4 the subset ∂F∗ of ∂F which is associated tothe smaller graph has Hausdorff dimension dimH(∂F∗) = D2. The inequality (1)implies that (18) remains true in the case a = 2.

We remark here that this method for determining the Hausdorff dimensioncan be always applied, provided that G(S) has a primitive subgraph with the samemaximal eigenvalue as G(S).

Of course, the approach of this paper is also applicable to JTCS, not associatedwith canonical number systems. As an example we consider a JTCS in R4. For aninteger p ≥ 4 let

B =

0 0 0 −p4

1 0 0 −p3

0 1 0 −p2

0 0 1 −p

, N = {(c, 0, 0, 0)t ∈ Z4 | 0 ≤ c < p4}.

To construct the graph G(S) we introduce the matrices

C :=

1 p p2 p3

0 1 p p2

0 0 1 p0 0 0 1

(BC)−1 =

0 1 −p 00 0 1 −p0 0 0 1−p−4 0 0 0

.


We first show that every m ∈ S is of the form m = Cm′ with m′ ∈ Z4: ByProposition 3.3 (i) and (iv) there is a k = (k1, k2, k3, k4) ∈ S and β ∈ Z with

|β| < p4 such that mβe1−−→ k. This implies k1 + β ≡ 0(mod 4) and C−1m =

(BC)−1(k+βe1) ∈ Z4. By the definition of G(Z4) we find for m = (m1,m2,m3,m4)and k = (k1, k2, k3, k4)

Cmβe1−−→ Ck ⇐⇒

kj = mj−1 (2 ≤ j ≤ 4)

k1 = −β −3∑i=0

pn−im4−i(19)

with |β| < p4. Let Cm ∈ S. Since S is a finite set, there is a constant α, such that|mj | ≤ α for 1 ≤ j ≤ 4. From (19) we obtain the bound

p4 ≥ |β| ≥ (p4 − p3 − p2 − p− 1)α,

Since p ≥ 4 we conclude that α < 2, and thus |mj | ≤ 1 for 1 ≤ j ≤ 4. The graphG(S) can easily be constructed by computer calculations. It has 30 vertices andis primitive. Moreover, in G(S0) each vertex can be reached from the vertex 0.By Lemma 3.6 this implies that (B,N ) is a JTCS. Hence, all necessary conditionsfor the application of Theorem 5.4 are fulfilled. This enables us to calculate theHausdroff dimension of ∂F and ensures the positivity and finiteness of the relatedHausdorff measure. For instance, we found in the case p = 4

dimH(∂F) = 3.690045932 . . . ,

and in the case p = 5

dimH(∂F) = 3.687372180 . . . .

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(Received: July 9, 1999)

W. Muller

Institut f. Statistik

Technische Universitat Graz

Lessingstr. 27

A–8010 Graz

Austria

E-mail: [email protected]

J. M. Thuswaldner

Institut f. Mathematik u. Statistik

Montanuniversitat Leoben

Franz-Josef-Str. 18

A–8700 Leoben

Austria


R. F. Tichy

Institut f. Mathematik

Technische Universitat Graz

Steyrergasse 30

A–8010 Graz

Austria


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