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CANONICAL NUMBER SYSTEMS, COUNTING AUTOMATA AND FRACTALS ¨ KLAUS SCHEICHER AND JORG M. THUSWALDNER Abstract. In this paper we study properties of the fundamental domain Fβ of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since Fβ defines a tiling of the n-dimensional vector space, we ask, which tiles of this tiling “touch” Fβ . It turns out, that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of Fβ . Since this boundary is a so-called graph-directed self affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

1. Introduction It is well known, that there exist many relations between number systems, automata and fractal sets. In this paper we focus on canonical number systems which are defined in algebraic number fields as well as in the n-dimensional real vector space. 1.1. Basic Definitions. First of all, let us define canonical number systems in number fields. Definition 1.1. Let K be a number field with ring of integers ZK . Let β ∈ ZK and N = {0, 1, . . . , |N (β)| − 1}, where N (β) denotes the norm of β over Q. The pair (β, N ) is called a canonical number system in K, if each γ ∈ ZK admits a unique representation of the form γ = η0 + η1 β + · · · + ηh β h with ηj ∈ N (0 ≤ j ≤ h) and ηh 6= 0 for h 6= 0. β is called base of this number system. It is a nontrivial problem to decide whether a given algebraic integer b can deserve as a base of a canonical number system or not. For quadratic number fields it was settled in a series of papers by Kátai, Kovács and Szabó (cf. [24, 25, 26]). The case of arbitrary number fields was treated in Kovács-Peth˝o [32]. In particular, they give an algorithmic characterization of the possible base numbers of canonical number systems in a given number field. Furthermore, they observed the following fact. If K is a number field, then Date: February 26, 2007. The first author was supported by the Austrian Science Foundation Project S8308. The second author was supported by the Austrian Science Foundation Project P-14200-MAT. 1

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¨ KLAUS SCHEICHER AND JORG M. THUSWALDNER

all but finitely many bases β of canonical number systems in K have the property, that their minimal polynomial mβ (x) = xn + bn−1 xn−1 + . . . + b1 x + b0 fulfills (1.1)

1 = bn < bn−1 < · · · < b1 < b0 .

In the present paper we will always assume, that this condition is fulfilled. It is needed in order to ensure, that certain properties are valid for a class of graphs, which will be discussed later on. We will need some easy facts about canonical number systems. Firstly, it is easy to see, that the Galois conjugates of a base β of a canonical number system have all modulus greater than 1 (cf. [32]). Furthermore, Kovács [31] proved, that if β is a base of a canonical number system in a number field K, then {1, β, . . . , β n−1 } forms an integral basis of K. This implies that canonical number systems can exist only in number fields having a power integral basis. The automata we will use, are so called “transducers” (cf. for instance Berstel [6] or Eilenberg [11]). Definition 1.2. The 6-tuple A = (Q, Σ, ∆, q, q0 , δ) is called a finite transducer automaton if • Q, Σ and ∆ are nonempty, finite sets, and • q : Q × Σ → Q and δ : Q × Σ → ∆ are unique mappings. The sets Σ and ∆ are called input and output alphabet, respectively. Q is the set of states and q0 is the starting state. The mappings q and δ are called transformation and result function, respectively. Next we define certain classes of sets, to which the fractals considered in the present paper belong to. Definition 1.3. Let G be a graph with vertices V labeled by {1, . . . , q} and edges E. Denote by Ei,j the set of edges leading from i to j. To each e ∈ E we associate a contraction mapping τe . Then one can show, that there exists a unique family of non-empty compact sets A1 , . . . , Aq , such that Ai :=

q [ [

τe (Aj )

(1 ≤ i ≤ q)

j=1 e∈Ei,j

(cf. Falconer [14, Chapter 3]). The set A := A1 ∪ . . . ∪ Aq is called a graph-directed set. If the contractions are similarities, A is called graph-directed self similar set, if they are affinities, it is called graph-directed self affine set. If q = 1 we just speak of a self similar and a self affine set, respectively. Recently, a special class of graph-directed self affine sets gained interest. The boundaries of so-called self affine tiles. We recall the definition of a tile (cf. for instance [40]).

CANONICAL NUMBER SYSTEMS, COUNTING AUTOMATA AND FRACTALS

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Definition 1.4. A self affine tile is a compact set T ⊂ Rn with nonempty interior that satisfies an identity of the shape [ MT = (T + d). d∈D

Here M is an expanding matrix (i.e. each of its eigenvalues has modulus greater than one), and D is a finite set of vectors in Rn , called the digit set, that satisfies |D| = | det M |. A tiling is a tesselation of Rn by tiles, whose interiors are pairwise disjoint. In the present paper we are concerned with a tiling, that is intimately related to canonical number systems in number fields. Namely, let (β, N ) be a canonical number system in a number field K with [K : Q] = n. Following Kátai-Környei [23] we define the fundamental domain Fβ related to (β, N ) in the following way. Let ¯ ) ( h ¯ X ¯ ck β −k , ck ∈ N . Eβ,h := x ∈ K ¯ x = ¯ j=1

Consider the embedding Φ:

n

K→R ,

n−1 X

αj β j 7→ (α0 , . . . , αn−1 ).

j=0

Define the matrix



0 ··· ··· ··· 1 . . .   . 0 . . . . . B :=   .. . . . ... ... . . .. ..  .. . . 0 ··· ··· 0 The fundamental domain of (β, N ) is now defined ( X Fβ := B −j aj | aj

0 .. . .. . .. .

−b0



−b1    −b2   ..  . .  ..  0 .  1 −bn−1 by ) ∈ Φ(N )

.

j≥1

It is shown in [23] that Fβ is a compact set that defines a tiling of the Rn . Moreover, F is a self affine set. If K is an imaginary quadratic field, it even turns out to be a self similar set, since in this case all conjugates of β have the same modulus. Because (1.2)

S

Φ(βx) = BΦ(x)

we have Fβ = h≥1 Φ(Eβ,h ). We are interested in the set of “neighbours” of Fβ in the tiling which is generated by Fβ . Therefore, we define the set S := {q ∈ Zn \ {0} | (Fβ ∩ Fβ + q) 6= ∅},

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which contains the translation vectors of all translates of Fβ by vectors with integral coordinates, which “touch” Fβ . S is a finite set, since Fβ is compact. Let et (1 ≤ t ≤ n) be the canonical basis vectors of Rn and define M := {me1 | m ∈ N }. Since {1, β, . . . , β n−1 } forms an integral basis of K we have Φ(ZK ) = Zn . Thus (B, M) forms a number system in Zn in the following sense. Each z ∈ Zn admits a unique “B-adic” representation of the form h X z= B j σj , j=0

where σj ∈ M (0 ≤ j ≤ h) and σh 6= 0 for h 6= 0. This number system is called the number system induced by the canonical number system (β, N ). By (1.2) (β, N ) and (B, M) have the same arithmetic behaviour. It will often be convenient for us, to work with (B, M) instead of (β, N ). 1.2. Aims of the Present Paper and Previous Results. In this paper we will prove the following results: • It is well-known, that one can construct transducers, that perform the addition of one in a canonical number system (cf. for instance Grabner et al. [18], Thuswaldner [42] or Scheicher [37]) In Section 2 we show, that this automaton is primitive, i.e. it is strongly connected and the lengths of its cycles have greatest common divisor 1. • In Section 3 we deal with the set S defined above. In Indlekofer et al. [21] and in a more general form in Strichartz-Wang [40] there are given algorithms for the determination of the set S. In each of them, one starts with a large graph, successively deleting vertices until one arrives at a graph, whose vertices correspond to the elements of S. The disadvantage of these algorithms is the fact that the graph, which forms the starting point of the deletion process can become very large. This is the case in particular if the eigenvalues of the base matrix B are close to one. In the present paper, we establish an algorithm, which starts from the graph, which performs the addition of one. The graph whose vertices correspond to the elements of S is obtained by considering certain products of this addition graph with itself. Since the addition graph is in general rather easy to determine (cf. for instance Gröchenig-Haas [19]), our algorithm is much easier to perform. We can even apply it in a forthcoming paper in order to determine S for a large class of canonical number systems (cf. Akiyama-Thuswaldner [3]). • In Section 4 the above-mentioned addition graph is used in order to determine the box counting dimension of the boundary of the self affine tile Fβ . This generalizes a result of Thuswaldner [43]. The methods we are using in this section are similar to that used in Deliu et al. [8]. However, in our case there occur certain overlaps. Furthermore, in [8] there are considered only two dimensional sets. There are many published papers which deal with the relations between number systems, automata and fractals. First of all, we mention Knuth’s book [30, p. 608] where the so called “twin dragon” is studied. This set turns out to be a special case of the tiles Fβ


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we are studying here, since it is the fundamental domain of the canonical number system (−1+i, {0, 1}) of the Gaussian number field Q(i). Note, that the twin dragon is self similar, because Q(i) is an imaginary quadratic number field and, hence, the conjugates −1 + i and −1 − i have the same modulus. Two other famous fractals occurring in connection with number systems are the Levy-dragon (cf. Duvall-Keesling [9]) and the Rauzy fractal (cf. Sirvent [38]). In Akiyama [1] and Akiyama-Sadahiro [2], among other things the fractal properties of number systems associated to Pisot numbers are discussed. Relations between Pisot number systems and automata have been considered extensively by Christiane Frougny and her co-writers (cf. for instance [15, 16] and the references given there) and in Thurston [41]. The fact that the addition automaton mentioned above can be used to describe the boundary of a tile was noted in Gröchenig-Haas [19], and earlier for a special case in Gilbert [17]. Indlekofer et al. [21] considered fundamental domains and automata related to certain quadratic number systems, which do not yield a representation for each number. This leads to totally disconnected fundamental domains. Kátai [22] defines socalled just touching covering systems, i.e. classes of number systems, which allow a tiling of the plane and gives certain characterizations of these objects. Tiles and tilings have been studied extensively in the last years (cf. for instance Vince [45] where a large list of literature is provided). In many cases the boundary of a tile has fractal structure. If the tile is a self affine set its boundary turns out to be a graph-directed self affine set, if it is a self similar set, its boundary is graph-directed self similar. The Hausdorff dimension of the boundary was calculated for different classes of self similar tiles by many authors (cf. for instance Bandt [4], Duvall et al. [10], M¨ uller et al. [34], Ngai-Wang [35], Strichartz-Wang [40], Kenyon et al. [28], Vince [45] and Wang [46], where also other interesting properties of tilings are studied). In these papers there are shown also many properties of self affine tilings. For self affine tiles the calculation of the dimension of the boundary becomes much more complicated. The Hausdorff dimension of a special class of graph-directed self affine sets has been calculated in Kenyon-Peres [29], but the fundamental domains discussed in the present paper do not belong to this class. Concerning the box counting dimension of graph directed self-affine tiles, there exists a paper by Deliu et. al. [8], where a general method for the determination of the box counting dimension of two dimensional graph-directed self-affine sets is presented. Similar methods as in [8] were used in Thuswaldner [43] in order to calculate the box counting dimension of the boundary of self affine tiles occurring in connection with number systems defined in real quadratic number fields. Moreover, we want to mention the paper of Veerman [44], where upper and lower bounds for the Hausdorff dimension of the boundary of a class of self affine tiles are given. Even in the simpler case of self affine sets only very special classes could be treated so far (cf. Lalley-Gatzouras [33] and H¨ uter-Lalley [20]). The problem of determining the Hausdorff dimension of self affine sets is strongly related to the behaviour of infinite Bernoulli convolutions (we refer to the excellent survey paper [36] by Peres, Schlag and Solomyak, where interesting references are given).


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2. Counting Automata for Canonical Number Systems In this section we want to discuss certain properties of the counting automaton, which performs the addition of 1 in the canonical number system (β, N ). These properties are of interest in their own right, but are also used in the following sections. For instance, in Duvall et al. [10] it was observed, that this automaton describes the boundary of the tile Fβ 1. This fact will be used in order to calculate the box counting dimension of the boundary of Fβ in Section 4. Some of the results of this section are published in Scheicher [37] and we will state them without proof. It is convenient for us to work in the number system (B, M) which is induced by the canonical number system (β, N ). Now we want to discuss briefly, how to construct the counting automaton A0 (1), which P j performs the addition of 1 in (B, M). We will use the notation (σN , . . . , σ0 ) = N j=0 B σj . n Let v ∈ Z . We want to add 1 to the B-adic representation v = (dN (v), dN −1 (v), . . . , d0 (v)), i.e. we want to construct the B-adic representation v + 1 = (dN 0 (v + 1), dN 0 −1 (v + 1), . . . , d0 (v + 1)). In analogy to the addition in “ordinary” q-adic number systems, we perform the addition digit wise. First we want to add 1 to the first digit d0 (v). This addition produces a carry q1 obeying the scheme d0 (v) + 1 = d0 (v + 1) + Bq1 . Note, that d0 (v +1) and q1 are determined uniquely by this scheme, since each v ∈ Zn has a unique B-adic representation. This reduces the problem of adding 1 to v to the problem of adding q1 to (dN (v), dN −1 (v), . . . , d1 (v)). Iterating this procedure yields the general scheme (2.1)

dj (v) + qj = dj (v + 1) + Bqj+1

(j ≥ 0).

Since v as well as v + 1 have finite B-adic representations, after finitely many, say J, steps we reach the constellation 0 + qj = 0 + Bqj+1 (j ≥ J). Since qJ also has a finite B-adic representation, this implies, that qj = 0 for j ≥ J. This algorithm can be performed with help of a transducer automaton. Adopting the notation of Definition 1.2 we define the counting automaton A0 (1) by setting Q = the set of possible carries, Σ = ∆ = M, q0 = 1, q : Q × Σ → Q : (qj , dj (v)) 7→ qj+1 according to (2.1), δ : Q × Σ → ∆ : (qj , dj (v)) 7→ dj (v + 1) according to (2.1). Note, that (2.1) defines the mappings q and δ uniquely. The finiteness of the automaton A0 (1) has been shown in Scheicher [37]. A0 (1) works in the following way. Starting at state 1, it reads the digits of the B-adic representation of v ∈ Zn from right to left. Suppose that A0 (1) rests at state qj and reads the digit dj (v). Then it moves to the state q(qj , dj (v)) and writes out the digit δ(qj , dj (v)). 1We

mention that in [10] the authors work with the so-called “contact matrix”. It is easy to see, that this contact matrix is the accompanying matrix of the counting automaton A(1), which we will define below.


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The output string (written from right to left) is the B-adic representation of v + 1. Since each v ∈ Zn has a finite B-adic representation, A0 (1) will end up in the state 0 after finitely many steps. Of course, the automaton A0 (1) can be visualized with help of a finite graph. The vertices of this graph are the elements of Q, its edges are defined in the following way. There exists an edge from q1 to q2 labeled by j1 |j2 , where j1 is the input digit and j2 is the output digit, if and only if j1 + q1 = j2 + Bq2 . We will identify this labeled, directed graph with the automaton A0 (1). It will be important later to have a good estimate for the walks of length k in A0 (1), which do not end up in the zero state. To this matter we need more information on A0 (1). First of all, we give a characterization of the set of states Q. This characterization is due to Scheicher [37]. Note, that we confine ourselves to canonical number systems whose minimal polynomial satisfies (1.1). For convenience we set bj = 0 for j > n. Lemma 2.1. Let A0 (1) be the automaton, which performs the addition of 1 in the number system (B, M). Then each state of A0 (1) can be considered as a carry occurring at the addition of 1. Such a carry has the shape X q= B j (bi1 +j − bi2 +j + bi3 +j − + · · · ) j≥0

with {i1 , i2 , . . .} a nonempty subset of {0, . . . , n}. Note that the above sum is finite. We shall use the notation q = [bi1 − bi2 + bi3 − . . .]. Let x=

X i≥0

B i σi

and x + 1 =

X

B i δi .

i≥0

According to the addition scheme (2.1) the move from a state q1 to a state q2 is described in the following way. Let q1 = [bi1 − bi2 + bi3 − . . .]. Then the following two cases can occur. (i) Let σi ∈ {0, . . . , b0 − (bi1 − bi2 + . . .) − 1}. In this case the actual carry is just worked off. We get δi = σi + bi1 − bi2 + . . . and q2 = [bi1 +1 − bi2 +1 + . . .]. (ii) Let σi ∈ {b0 − (bi1 − bi2 + . . .), . . . , b0 − 1}. In this case an additional carry occurs. We get δi = σi − b0 + bi1 − bi2 + . . . and q2 = [b0 − b1 + bi1 +1 − bi2 +1 + . . .]. We will need this characterization of the set of states in order to prove certain properties of A0 (1). Let A(1) be the graph emerging from A0 (1) if one omits the state 0 and each edge leading to it. Then A(1) has the following properties. Proposition 2.2. The automaton A(1) is strongly connected, i.e. if q1 and q2 are arbitrary states of A(1), then there exists a walk from q1 to q2 . Proof. First we construct a walk from q1 to [bn ] = 1. It is easily seen, that this is done by applying rule (i) sufficiently often, i.e. to use an input string consisting of sufficiently many zeros. It remains to construct a way from 1 to q2 . Such a way exists trivially, because A(1) consists of all the possible nonzero carries occurring at the addition of 1. If there would be


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no way from 1 to q2 , this would imply, that q2 is no possible carry. But if q2 is no possible carry, it does not occur in the set of states, which is a contradiction. ¤ Proposition 2.3. For each 2 ≤ p ≤ n there exists a cycle of length p in A(1). Proof. In order to prove the present result we will construct a cycle of length p in A(1). We start at state " # X c0 = [b0 − b1 + bp − bp+1 + b2p − b2p+1 + . . .] = (bpj − bpj+1 ) . j≥0

Note, that this sum is finite, because bk = 0 for k > n. Now suppose, that the input string has the shape (b0 − 1, 0, . . . , 0 ). According to the above rules for moving through A(1) | {z } p−1 times we will pass the following states " # X c0 = (bpk − bpk+1 ) , k≥0

c1 ... cp−1

# " X (bpk+1 − bpk+2 ) , = k≥0

.". . # X (bpk+p−1 − bpk+p ) , = "

cp =

k≥0

b0 − b1 +

X k≥0

# " X (bpk − bpk+1 ) = c0 . (bp(k+1) − bp(k+1)+1 ) = #

k≥0

Note, that we used p − 1 times rule (i) for the steps from c0 to cp−1 . For the last step from cp−1 to cp we used rule (ii). Since 2 ≤ p ≤ n and bk < bk+1 , each of the states cj is different from the state [b0 ] = 0 which was excluded in the definition of A(1). This proves the existence of a cycle of length p for p in the indicated range. ¤ Let G be a directed graph. It is well known, that if G is strongly connected and the greatest common divisor of the length of its closed directed walks is 1, then G is primitive. By the Perron-Frobenius Theorem (cf. Brualdi-Ryser [7, p. 68]) the accompanying matrix of a primitive graph has a unique (real, positive) largest eigenvalue. With this definition Proposition 2.2 and Proposition 2.3 immediately imply the following result. Theorem 2.4. Let (β, N ) be a canonical number system whose base β fulfills the monotonicity condition (1.1). Then the graph A(1) is a primitive graph. Proof. For n ≥ 3 A(1) has cycles of length 2 and 3 and the corollary follows since gcd(2, 3) = 1. For n = 2 A(1) is known explicitly (cf. Thuswaldner [42]) and it contains cycles of length 2 and 3. ¤


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In the remaining part of the paper we will often have to work with certain sets of walks of a finite graph G. To this matter we will need the following abbreviations. Let P (G) be the set of all walks in G and define Pk (G) := {w ∈ P (G) | w has length k}. With these notations we get the following corollary. Corollary 2.5. Let P be the accompanying matrix of the graph A(1). Then P has a unique (positive, real) largest eigenvalue µ < |N (β)| and there exists a constant c such that |Pk (A(1))| = cµk (1 + o(1)). Proof. This follows from Theorem 2.4 (cf. Gröchenig-Haas [19, Theorems 2.1 and 4.8]). ¤ In the next section we are concerned with the set S. In order to give an algorithm for its determination, we need a certain product of A0 (1) with itself. We will now define this product automaton. Consider the automaton A0 (1). If this automaton processes the B-adic representation of a certain u ∈ Zn starting at state q0 = 1, its output string is the representation of u + 1. Of course, we could take any state q of A0 (1) as starting state. Processing u with starting state q will result in the output string u + q = (dN (u + q), . . . , d0 (u + q)). Let u ∈ Zn . Then with help of A0 (1) one can produce the B-adic representation of u + q1 for each q1 ∈ Q. If we take the output string and use it again as input string for a second copy of A0 (1) with starting state q2 ∈ Q, we can produce the B-adic representations of u + q1 + q2 for q1 , q2 ∈ Q. Scheicher [37] established the following result. Lemma 2.6. Any v ∈ Zn can be added with help of finitely many applications of A0 (1). With m applications one can add any vector contained in the set ( m ) X ¯¯ mQ = qj ¯ qj ∈ Q . j=1

Proof. Cf. Scheicher [37, p. 5].

¤

Suppose that we want to add a vector v ∈ mQ. By the definition of the set mQ there P (j) (j) (1) (m) exist q0 ∈ Q (1 ≤ j ≤ m) with v = m j=1 q0 . Consider m copies A0 (1), . . . , A0 (1) of (j)

(j)

the automaton A0 (1), where the j-th copy A0 (1) rests in state q0 . To add v ∈ mQ to the B-adic representation of u ∈ Zn , we proceed as follows. We put the first digit of u in (1) (1) (2) A0 (1), the output digit of A0 (1) is used as input digit for A0 (1), and so on until we (m) (j) arrive at the output digit of A0 (1). After this procedure the j-th copy A0 (1) rests in (j) state q1 . Processing all the other digits (including leading zeros) in the same way results in an output string, that is equal to the B-adic representation of u + v. We can regard the m copies of A0 (1) as a product automaton A0 (m). To this matter we define the following product. Let A1 and A01 be counting automata. Then the product automaton A2 := A1 ⊗ A01 is defined in the following way Let a1 , b1 be states of A1 and a01 , b01 be states of A01 . Furthermore, let `1 , `01 , `2 ∈ M.


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• a2 is a state of A2 if a2 = a1 + a01 `1 |`2

• There exists an edge a2 −−→ b2 in A2 if there exist `1 |`0

`0 |`2

1 a1 −−→ b1 ∈ P1 (A1 ) and a01 −1−→ b01 ∈ P1 (A01 )

with a1 + a01 = a2 and b1 + b01 = b2 or there exist `1 |`0

`0 |`2

1 b01 ∈ P1 (A01 ) a1 −1−→ b1 ∈ P1 (A1 ) and a01 −−→

with a1 + a01 = a2 and b1 + b01 = b2 . Keeping in mind the above observations it is now easy to see, that (1)

(j)

(m)

A0 (m) = A0 (1) ⊗ · · · ⊗ A0 (1) = ⊗m j=1 A0 (1) is the automaton that can add the numbers v ∈ mQ. Since A0 (m) is a counting automaton σi |δi

there exists an edge qi −−→ qi+1 in A0 (m) if and only if (2.2)

qi + σi = Bqi+1 + δi .

The values qi can be considered as a carry occurring at the addition of a number of mQ. In the next section we will be interested only in states of A(m), which are the endpoint of a walk of infinite length. Thus for an automaton A we denote by Red(A) the automaton that emerges from A, if all states of A, which are not the endpoint of a walk of infinite length, are removed. In the next lemma we show, that in computing A0 (m) reducing after each step of the multiplication process is the same as reducing at the end of the multiplication process. Lemma 2.7. We have the identity (1)

(2)

(3)

(m)

(2.3) Red(A0 (m)) = Red(. . . Red(Red(Red(A0 (1)) ⊗ A0 (1)) ⊗ A0 (1)) . . . ⊗ A0 (1)). Proof. We prove this lemma by induction on m. For m = 1 there is nothing to prove. Thus we assume, that (2.3) is true for m − 1 instead of m. Using this assumption, (2.3) becomes (2.4)

(m)

Red(A0 (m)) = Red(Red(A0 (m − 1)) ⊗ A0 (1)).

The lemma will be proved, if we can show that (2.4) holds. Since (m)

(2.5)

Red(A0 (m)) = Red(A0 (m − 1) ⊗ A0 (1)) (m)

we conclude that Red(A0 (m)) ⊃ Red(Red(A0 (m−1))⊗A0 (1)). It remains to establish the `1 |`2

opposite inclusion in order to prove (2.4). Suppose that rm1 −−→ rm2 ∈ P1 (Red(A0 (m)). By (2.5) there exist (2.6)

`1 |`0

`0 |`2

1 rm−1,1 −−→ rm−1,2 ∈ P1 (A0 (m − 1)) and r11 −1−→ r12 ∈ P1 (A0 (1))

with r11 + rm−1,1 = rm1 and r12 + rm−1,2 = rm2 or there exist `0 |`2

`1 |`0

1 r12 ∈ P1 (A0 (1)) rm−1,1 −1−→ rm−1,2 ∈ P1 (A0 (m − 1)) and r11 −−→


11

with r11 + rm−1,1 = rm1 and r12 + rm−1,2 = rm2 . W.l.o.g. we suppose that the pair of edges in (2.6) exists. Since A0 (1) is strongly connected, r11 has infinitely many predecessors. Now we have to distinguish two cases. Case 1: rm−1,1 has infinitely many predecessors. Thus rm−1,1 ∈ Red(A0 (m − 1)) and, hence, rm−1,1 + r11 ∈ Red(A0 (m − 1)) ⊗ A0 (1). Since rm−1,1 + r11 = rm1 has infinitely many `1 |`2

predecessors in A0 (m) we conclude that rm1 −−→ rm2 is contained in Red(Red(A0 (m − 1)) ⊗ A0 (1)) and we are done. Case 2: rm−1,1 has only finitely many predecessors. Since rm1 has infinitely many predecessors by the definition of ⊗ there exist a state sm−1,1 of A0 (m − 1) and a state s11 of A0 (1) with rm1 = sm−1,1 + s11 such that sm−1,1 has infinitely many predecessors. Indeed, let N be the number of states in A0 (m − 1). Then there exists a walk dN |d0

d0 |d0

N 0 rm,−N −−−→ · · · −−−→ rm1

in A0 (m). Thus by the definition of ⊗ and since A0 (m − 1) represents the full shift there exist the walks dN |d00 d0 |d00 0 N sm−1,−N −−−→ · · · −−−→ sm−1,1 in A0 (m − 1) and d00 |d0

d00 |d0

0 0 N N −→ s11 −→ · · · −− s1,−N −− in A0 (1) such that rm1 = sm−1,1 +s11 . The pumping lemma implies that sm−1,1 has infinitely many predecessors. (2.6) now implies that

(2.7)

`1 |`00

`00 |`2

1 sm−1,1 −−→ sm−1,2 ∈ P1 (Red(A0 (m − 1))) and s11 −1−→ s12 ∈ P1 (A0 (1))

with s12 + sm−1,2 = rm2 or there exist (2.8)

`00 |`2

`1 |`00

1 sm−1,1 −1−→ sm−1,2 ∈ P1 (Red(A0 (m − 1))) and s11 −−→ s12 ∈ P1 (A0 (1))

with s12 + sm−1,2 = rm2 . Note, that the existence of the edges with the indicated labeling is due to the following observation. Let u ∈ Zn . If we compute the B-adic representation of u + rm−1,1 + r11 the result is the same as if we compute the B-adic representation of u + sm−1,1 + s11 , because rm−1,1 + r11 = sm−1,1 + s11 . (2.7) and (2.8) now imply that `1 |`2

rm1 −−→ rm2 ∈ Red(Red(A0 (m − 1)) ⊗ A0 (1)) and we are done.

¤

3. The Boundary of Fβ In this section we give an algorithm for the construction of a labeled graph, whose vertices correspond to the elements of S. This algorithm turns out to be much easier to perform than the known algorithms of [21] and [40]. With this algorithm it is easy to determine the set S for a given canonical number system. In the forthcoming paper [3] we can even apply it in order to determine the set S for a large class of canonical number systems. Furthermore, we state a result due to Duvall et al. [10], which describes the graph-directed self affinity of ∂Fβ and prove some projection conditions that will be needed for the dimension calculations in the last section.


12

Let (B, M) be the number system in Zn induced by the canonical number system (β, N ) and set Cq := Fβ ∩ (Fβ + q) (q ∈ Q). Lemma 3.1. ∂Fβ =

[

Cq .

q∈Q

Proof. We refer to [10] or [45].

¤

Let B(Zn ) be a labeled directed graph with set of vertices Zn . The labeled edges connecting two vertices are defined as follows. Let q1 , q2 ∈ Zn be two vertices of B(Zn ). Then there exists an edge from q1 to q2 labeled by j1 |j2 (j1 , j2 ∈ M) if and only if (3.1)

q1 + j1 = Bq2 + j2 .

Let B(S) be the restriction of B(Zn ) to the set S. Remark 3.2. Note, that in [34] this graph is defined in a slightly different way. Namely, its edges are directed in the opposite direction. We use the direction indicated in (3.1) to be able to work out the resemblance between B(S) and the counting automata defined in the previous section. All the results we need from [34] remain valid also for this slightly different setting. Remark 3.3. There are also different methods to construct the boundary of a tile. We refer to Kenyon [27], Vince [45, Section 6] and Sirvent-Wang [39], where so-called Rauzy fractals are constructed with help of substitutions. Furthermore, we want to mention that the accompanying matrix of B(S) is used in Wang [46] in order to compute the Hausdorff dimension of graph-directed self similar sets which are the boundaries of certain tiles. Lemma 3.4. (cf. [34]) B(S) is the union of all cycles of B(Zn ) apart from the cycle at 0, and all walks of B(Zn ), which connect two of these cycles. This implies, that each vertex of B(S) is the end point of an infinite walk. By Lemma 2.6 each v ∈ Zn can be added by finitely many applications of A0 (1), i.e. there exists an m ∈ N such that v ∈ mQ. Set m0 := min{m ∈ N | S ⊂ mQ}. This implies that each q ∈ S is a state of A(m0 ). Since the states of A(m0 ) and B(S) are both elements of a subset of Zn and their edges are defined in the same way by (2.2) and (3.1) we have (3.2)

B(S) ⊂ A(m0 ).

Theorem 3.5. The graph B(S), and with it the set S, can be determined by the following algorithm: m := 1 A[1] := A0 (1) repeat m := m + 1 A[m] := Red(A[m − 1] ⊗ A[1])


13

until A[m] = A[m − 1] B(S) := A[m] \ {0} This algorithm always terminates after finitely many steps. Proof. We first show that the algorithm terminates. By Lemma 3.4 the inclusion (3.2) implies B(S) ⊂ Red(A0 (m0 )). Furthermore, Lemma 2.7 yields Red(A0 (m0 )) = A[m0 ]. Since B(S) does not contain the vertex 0 we arrive at (3.3)

B(S) ⊂ A[m0 ] \ {0}.

But (3.3) implies together with Lemma 3.4 that A[m0 ] contains each reduced finite subgraph of B(Zn ) having no stranded vertex. In particular, A[m0 + 1] ⊂ A[m0 ], and since the opposite inclusion is trivial we conclude that equality holds. Thus the algorithm terminates for an m1 ≤ m0 + 1. Because 0 has no successors in A[m0 ] we have A[m0 ] \ {0} = Red(A[m0 ] \ {0}). Since by Lemma 3.4 B(S) contains each reduced finite subgraph of B(Zn \ {0}) having no stranded vertex we conclude that (3.4)

B(S) ⊃ A[m0 ] \ {0}.

It is easy to see, that A[m] = A[m+1] implies A[m] = A[m+d] for each d ∈ N. In particular, we have A[m1 ] = A[m0 ]. Together with (3.3) and (3.4) this shows that B(S) = A[m1 ]\{0}. and thus the algorithm yields B(S). ¤ It was shown for instance in Duvall et al. [10] that ∂Fβ is a graph-directed self affine set directed by the graph A(1). In particular we have the following lemma. Lemma 3.6. For all states q ∈ Q \ {0} of A(1) we have [ Cq = ϕa (Cr ) a,r a|a0

r− −→q

with ϕa (x) = B −1 (x + a). This implies together with Lemma 3.1, that [ [ ∂Fβ = ϕa (Cr ). q∈Q\{0}

a,r a|a0

r− −→q a|a0

Note, that the unions over a, r run over all edges r −−→ q in A(1) leading to a fixed state q. Proof. Cf. [10, Section 3] and note, that the contact matrix used there is exactly the accompanying matrix of A(1). ¤ In order to simplify the dimension calculations of the following section we show that we can work with a real diagonal matrix Λ instead of the matrix B. Let β = β1 , . . . , βs+2t with βs+2j−1 = β¯s+2j be the Galois conjugates of β (n = s + 2t). Then there exists a regular real matrix T such that T BT −1 = Λ with Λ := diag(β1 , . . . , βs , Σs+1 , . . . , Σs+t )


14

where

µ Σs+j :=

0 for 1 ≤ j ≤ t.


15

Then for any W ∈ Pk (A(1)) λn−1 (Πj (ψW (Dq ))) > 0 λn−2 (Πs+j (ψW (Dq ))) > 0

(1 ≤ j ≤ s), (1 ≤ j ≤ t)

holds. Proof. Define the quantities n Y (3.7) ξj := |β` | (1 ≤ j ≤ s),

ξs+j :=

`=1 `6=j

n Y

|β` | (1 ≤ j ≤ t).

`=1 `6=s+2j−1, s+2j

By the definition of ψW and the shape of the matrix Λ we have for 1 ≤ j ≤ s λn−1 (Πj (ψW (Dq ))) = λn−1 (Πj (Λ−k Dq )) = λn−1 (Λ−k Πj (Dq )) = ξj−k λn−1 (Πj (Dq )) > 0. By analogous reasoning one gets the claim for s + 1 ≤ j ≤ s + t.

¤

Proposition 3.8. Let q ∈ Q \ {0}. Then λn−1 (Πj (Dq )) > 0 (1 ≤ j ≤ s) and

λn−2 (Πs+j (Dq )) > 0 (1 ≤ j ≤ t).

Proof. We confine ourselves to the case 1 ≤ j ≤ s. The case s + 1 ≤ j ≤ s + t can be treated in the same way. Kátai-K˝ornyei [23] proved that Fβ contains an open ball. Since (n−1) (n−1) T is a regular matrix, the same holds for ∆. Thus Πj (∂∆) ⊃ Kδ , where Kδ is an (n − 1)-dimensional open ball. This implies, that   [ λn−1  Πj (Dr ) = λn−1 (Πj (∂∆)) > 0. r∈Q\{0}

Since the union is finite, there exists an r0 ∈ Q \ {0} with λn−1 (Πj (Dr0 )) > 0. Since A(1) is strongly connected, for each q ∈ Q \ {0} there exists a walk W leading from r0 to q. Thus ψW (Cr0 ) ⊂ Cq . Now it follows from Lemma 3.7, that λn−1 (Πj (Dq )) > 0 for all q ∈ Q \ {0}. ¤ Remark 3.9. In his papers [12, 13] Falconer observed that the positivity of the Lebesgue measure of certain projections of self affine fractals is important for establishing formulas for their box counting dimension. This was generalized in Deliu et al. [8]. Also in our case Proposition 3.8 turns out to be the key for the calculation of the box counting dimension of ∂Fβ . 4. Dimension Calculations In this section we calculate the box counting dimension of ∂Fβ . We will follow essentially the approach of Deliu et al. [8] and Thuswaldner [43], which calculated the box counting dimension of a class of two dimensional graph-directed self affine sets. Here we have to treat n-dimensional sets, which makes the calculations more involved. Moreover, there occur certain overlaps, which do not occur for the sets considered in [8].


16

We will need the following notations. Denote by bxc the greatest integer being less than or equal to x, let ξj for 1 ≤ j ≤ s + t be as in (3.7) and set βmax := max {|β` |}, 1≤`≤n

ζ (k) :=

n Y `=1

$µ

βmax |β` |

¶k % ,

(k) ζj

$µ ¶k % n Y βmax := |β` | `=1

(1 ≤ j ≤ s).

`6=j

Let W ∈ P (A(1)) be a walk with labeling (a1 , . . . , ak ). Then (3.5) implies, that Ã ! k X (4.1) ψW (x) = Λ−k x − Λ`−1 T a` . `=1

Let Y be an n-dimensional cube with side length L containing ∂∆ and fix k ∈ N. Iterating (3.6) k-times implies that the collection (1)

Ak := {ψW (Y ) | W ∈ Pk (A(1))} (1)

forms a cover of ∂∆. By (4.1), each of the elements of Ak is equal to a set of the form Λ−k (Y + T m) (m ∈ Zn ). Furthermore, the following result holds. Lemma 4.1. (cf. [43, Lemma 2.4]) There are at most |Q| different walks in Pk (A(1)) having the same labeling. They start in pairwise different states. After these preparations we can easily prove the following lemma (cf. Thuswaldner [43]). (1)

(1)

Lemma 4.2. Each set Z ∈ Ak intersects at most c1 sets of the collection Ak , where c1 is an absolute constant. Proof. Note, that Z is of the form Λ−k (Y + T v) for a certain v ∈ Zn . Thus Z can intersect only finitely many, say c01 , elements of the collection K := {Λ−k (Y + T v) | v ∈ Zn }. Let W, W 0 be two walks of A(1). Then the sets ψW (Y ) and ψW 0 (Y ) are equal, if and only if W and W 0 have the same labeling. Since this can be the case for at most |Q| different (1) walks by Lemma 4.1, the collection Ak contains each set of K at most |Q| times. Thus the result holds with c1 := |Q|c01 . ¤ (1)

The collection Ak has the disadvantage, that the ratio of the side lengths of its elements turn to 0 and ∞, respectively, for k → ∞. Thus we want to construct another collection. (1) To this matter consider a typical rectangular solid Z ∈ Ak . Let e1 , . . . , en be the canonical basis vectors of Rn and define fj := ej (1 ≤ j ≤ s), fs+2j−1 := µ2j−1 e2j−1 + µ2j e2j (1 ≤ j ≤ t), fs+2j := −µ2j e2j−1 + µ2j−1 e2j (1 ≤ j ≤ t)


17

for certain numbers µj ∈ R with µ22j−1 + µ22j = 1. It is easy to see that the fj form an orthonormal basis of Rn . With respect to this basis Z can be expressed easily (with µj (1 ≤ j ≤ 2t) selected properly). In fact there exists (x1 , . . . , xn ) ∈ Rn such that ( ) n X ¯ Z= v= αj fj ¯ 0 ≤ αj − xj ≤ L|βj |−k . j=1

Now we define a grid Γ by the fundamental mesh ) ( ¯ n j k−1 X ¯ . (4.2) M := v = αj fj ¯¯ 0 ≤ αj − xj ≤ L|βj |−k (βmax /|βj |)k j=1

This grid subdivides Z into ζ (k) rectangular solids. Call the set of these solids M(Z). These solids have the advantage, that the ratios of their side lengths are between 12 and 2. So, they are “almost” n-dimensional cubes. We now define the collections n o (2) (1) Ak := M | M ∈ M(Z) for some Z ∈ Ak (k ∈ N). Apart from these collections we will need the collections n o (∩) (2) Ak := M ∈ Ak |M ∩ ∂∆ 6= ∅

(k ∈ N).

(∩)

(∩)

Our goal is to estimate the elements of Ak . We will do this by comparing Ak collections defined above. It is easy to prove the following estimate.

with the

Lemma 4.3. There exist absolute constants c1 , c2 > 0, such that (2)

c1 µk ζ (k) ≤ |Ak | ≤ c2 µk ζ (k) . (1)

Proof. The collections Ak contain exactly |Pk (A(1))| elements. By Corollary 2.5 we get (1) (1) the estimate c1 µk ≤ |Ak | ≤ c2 µk . Since to each element of Ak there correspond exactly (2) (2) ζ (k) elements of Ak , the estimates for Ak follow. ¤ (2)

(∩)

We want to compare Ak with Ak . In this step we make use of the projection-results of the preceding section. Lemma 4.4. There exists an absolute constant c3 such that (2)

(∩)

(2)

c3 |Ak | ≤ |Ak | ≤ |Ak |. Proof. The second inequality is trivial, thus we have to prove only the first one. Let j0 be the (smallest) index, for that |βj0 | = βmax . First we prove the result for 1 ≤ j0 ≤ s, i.e. for the case, where one of the eigenvalues of greatest modulus is real. (1) It is clear from above that to each Z ∈ Ak there correspond exactly ζ (k) elements (2) (∩) (1) of Ak . We want to estimate, how many M ∈ Ak correspond to a given Z ∈ Ak . We do this by comparing the Lebesgue measure of Πj0 (M ) with the Lebesgue measure of Πj0 (∂∆ ∩ Z).


18

(2)

Let M ∈ M(Z) be an element of Ak . It is easy to see, that (4.3)

λn−1 (Πj0 (M )) =

Ln−1 (k)

ξjk0 ζj0

=

Ln−1 . ξjk0 ζ (k)

The second equality follows, because |βj0 | = βmax . On the other hand, for a certain q ∈ Q \ {0} there exists a walk W ∈ Pk (A(1)) starting at q, such that Z = ψW (Y ). Since Dq ⊂ Y we conclude, that ψW (Dq ) ⊂ ∂∆ ∩ Z. We want to estimate the Lebesgue measure of Πj0 (∂∆ ∩ Z). An application of Proposition 3.8 yields (4.4) c4 λn−1 (Πj0 (∂∆∩Z)) ≥ λn−1 (Πj0 (ψW (Dq ))) = λn−1 (ψW (Πj0 (Dq ))) = ξj−k λn−1 (Πj0 (Dq )) = k 0 ξj0 with some constant c4 > 0. ¥ ¦ Comparing (4.3) and (4.4) yields, that the Πj0 -projections of at least gk := c4 ζ (k) /Ln−1 (1) elements of M(Z) are necessary to cover Πj0 (∂∆ ∩ Z). Thus for each Z ∈ Ak there exist (2) at least gk elements of Ak having nonempty intersection with ∂∆. But this implies, that (∩) (1) |Ak | ≥ gk |Ak | and the result follows with c3 = c4 /2Ln−1 . If the eigenvalue of largest modulus is complex, the arguments are very similar to the real case, and we omit the details. ¤ By combining Lemmas 4.3 and 4.4 we derive the following result. (∩)

Proposition 4.5. The elements of Ak yield a cover of ∂∆. Moreover, there exist positive constants c5 , c6 such that (∩)

c5 µk ζ (k) ≤ |Ak | ≤ c6 µk ζ (k) . (∩)

Before we can prove our main result, we have to ensure, that the elements of Ak not overlap too much. In particular, we show the following lemma.

do (∩)

Lemma 4.6. Each point of the Rn is covered at most c7 times by the elements of Ak , where c7 > 0 is an absolute constant. (∩)

Proof. It is an easy consequence of Lemma 4.2, that an element W ∈ Ak can intersect at (1) most c1 elements of Ak . In each of the c1 elements there are contained at most 3n elements (∩) of Ak having nonempty intersection with W . The result follows with c7 := 3n c1 . ¤ After these preparations we are able to compute the box counting dimension of ∂Fβ . We recall the definition of the box counting dimension of a set E ⊂ Rn (cf. for instance Falconer [14, p. 20]). Let Nr (E) be the number of n-dimensional r-mesh cubes that intersect E. Then the box counting dimension is defined by log Nr (E) , r→0 − log r

dimB E = lim


19

provided, that the limit exists. By Barnsley [5, p. 176, Theorem 1], this is equivalent to log Nrk (E) , k→∞ − log rk

dimB E = lim

where rk := c8 rk , with some fixed r ∈ (0, 1) and c8 > 0. Theorem 4.7. Let (β, N ) be a canonical number system, and suppose that the coefficients of the minimal polynomial mβ (x) = xn + bn−1 xn−1 + · · · + b1 x + b0 fulfill (1.1). Then the box counting dimension of the boundary of its fundamental domain Fβ is given by log µ + n log βmax − log |N (β)| . log βmax Here βmax := max1≤j≤n {|βj |} is the maximum of the moduli of all Galois conjugates of β, and µ is the unique (real, positive) largest eigenvalue of the accompanying matrix of the automaton A(1). dimB ∂Fβ =

Proof. We calculate the box counting dimension of ∂∆. Since T is a regular matrix, −k dimB ∂∆ = dimB T ∂Fβ = dimB ∂Fβ . For k ∈ N let Ck be the set of n-dimensional Lβmax mesh cubes that intersect ∂∆. First we show, that the chain of inequalities (∩)

(4.5)

c9 |Ck | ≤ |Ak | ≤ c10 |Ck |

holds for certain absolute constants c9 , c10 > 0. (∩) −k Since all side lengths of each √ rectangular solid M ∈ Ak are smaller than 2Lβmax ,M n can not intersect more than (4 n) elements of Ck . Since each X ∈ Ck has nonempty (∩) intersection with at least one M ∈ Ak , this gives the first inequality of (4.5) with c9 := √ −n (4 n) . (∩) Let X ∈ Ck and M ∈ Ak . Then X ∩ M 6= ∅ implies, that M ⊂ X5 , where X5 denotes the n-dimensional cube with side length five times as large as the side length of X centered −nk at X. The content of X5 is 5n Ln βmax . On the other hand, it follows from (4.2) that the −nk content of M is greater than Ln βmax . Suppose now, that there exist c10 := 5n (c7 + 1) (∩) elements of Ak having nonempty intersection with X. Each of these c10 elements has to be a subset of X5 . Comparing volumes yields, that there has to be at least one point in (∩) X5 , that is covered by at least c7 + 1 elements of Ak . This contradicts Lemma 4.6. So (∩) to each X ∈ Ck there correspond less than c10 elements of Ak . Since by the construction (∩) (∩) of the collection Ak each M ∈ Ak has nonempty intersection with at least one X ∈ Ck , the second inequality of (4.5) follows. Now, applying (4.5) and Proposition 4.5 yields (∩)

log |Ak | log(µk ζ (k) ) log |Ck | = lim = lim . dimB ∂∆ = lim −k ) −k ) −k ) k→∞ − log(Lβmax k→∞ − log(Lβmax k→∞ − log(Lβmax Since ζ

(k)

=

¶k n µ Y βmax j=1

|βj |

(1 + o(1)) =

nk βmax (1 + o(1)) |N (β)|k


20

we derive

³

dimB ∂∆ = lim

log

k→∞

nk µk βmax (1 |N (β)|k

+ o(1))k

k log βmax

´ =

log µ + n log βmax − log |N (β)| . log βmax

This proves the theorem.

¤ References

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