TOPOLOGICAL PROPERTIES OF TWO-DIMENSIONAL NUMBER SYSTEMS

SHIGEKI AKIYAMA AND JÖRG M. THUSWALDNER

Abstract. In the two dimensional real vector space ℝ^{2} one can define analogs of the well-known q-adic number systems. In these number systems a matrix M plays the role of the base number q. In the present paper we study the so-called fundamental domain F of such number systems. This is the set of all elements of ℝ^{2} having zero integer part in their “M-adic” representation. It was proved by Kátai and Kőrnyei, that F is a compact set and certain translates of it form a tiling of the ℝ^{2}. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of F and give a result on the structure of its inner points.

1. Introduction

In this paper we use the following notations: ℝ, ℚ, ℤ and ℕ denote the set of real numbers, rational numbers, integers and positive integers, respectively. If x ∈ ℝ we will write ⌊x⌋ for the largest integer less than or equal to x. λ will denote the 2-dimensional Lebesgue measure. Furthermore, we write ∂A for the boundary of the set A and int(A) for its interior. diag(λ_{1}, λ_{2}) denotes a 2 × 2 diagonal matrix with diagonal elements λ_{1} and λ_{2}.

Let q ≥ 2 be an integer. Then each positive integer n has a unique q-adic representation of the shape n = ∑_{k=0}^{H} a_{k}q^{k} with a_{k} ∈ {0, 1, …, q − 1} (0 ≤ k ≤ H) and a_{H} ≠ 0 for H ≠ 0. These q-adic number systems have been generalized in various ways. In the present paper we deal with analogs of these number systems in the 2-dimensional real vector space, that emerge from number systems in quadratic number fields. The first major step in the investigation of number systems in number fields was done by Knuth [13], who studied number systems with negative bases as well as number systems in the ring of Gaussian integers. Meanwhile, Kátai, Kovács, Pethő and Szabó invented a general notion of number systems in rings of integers of number fields, the so-called canonical number systems (cf. for instance [10, 11, 12, 15]). We recall their definition.

Let K be a number field with ring of integers Z_{K}. For an algebraic integer b ∈ Z_{K} define 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 if any γ ∈ Z_{K} admits a representation of the shape

γ = c_{0} + c_{1}b + ⋯ + c_{H}b^{H},

where c_{k} ∈ N (1 ≤ k ≤ H) and c_{H} ≠ 0 for H ≠ 0.

Date: June 11, 1999.

1991 Mathematics Subject Classification. 11A63.

Key words and phrases. Radix representation, Connectedness.

2 SHIGEKI AKIYAMA AND JÖRG M. THUSWALDNER

These number systems resemble a natural generalization of q-adic number systems to number fields. Each of these number systems gives rise to a number system in the ndimensional real vector space. Since we are only interested in the 2-dimensional case, we construct these number systems only for this case. Consider a canonical number system (b, N) in a quadratic number field K with ring of integers Z_{K}. Let p_{b}(x) = x^{2} + Ax + B be the minimal polynomial of b. It is known, that for bases of canonical number systems −1 ≤ A ≤ B ≥ 2 holds (cf. [10, 11, 12]). Now consider the embedding Φ : K → ℝ^{2}, α_{1} + α_{2}b ↦ (α_{1}, α_{2}), where α_{1}, α_{2} ∈ ℚ. Kovacs [14] proved, that {1, b} forms an integral basis of Z_{K}. Thus we have Φ(Z_{K}) = ℤ^{2}. Furthermore, note that Φ(bz) = MΦ(z) with

M = (\begin{matrix}0 & −B \\ 1 & −A \end{matrix}) .

Since the elements of N are rational integers, for each c ∈ N, Φ(c) = (c, 0)^{T}. Summing up we see, that (M, Φ(N)) forms a number system in the two dimensional real vector space in the following sense (cf. also [8], where some properties of these number systems are studied). Each g ∈ ℤ^{2} has a unique representation of the form

g = d_{0} + Md_{1} + … + M^{H}d_{H},

with d_{k} ∈ Φ(N) (1 ≤ k ≤ H) and d_{H} ≠ (0, 0)^{T} for H ≠ 0. These number systems form the object of this paper. In particular, we want to study the so-called fundamental domains of these number systems. The fundamental domain of a number system (M, Φ(N)) is defined by

F = {z | z = ∑_{j≥1} M^{−j}d_{j}, d_{j} ∈ Φ(N)} .

Sloppily spoken, F contains all elements of ℝ^{2}, with integer part zero in their “M-adic” representation. In Figure 1 the fundamental domain corresponding to the M-adic representations arising from the Gaussian integer −1 + i is depicted. This so-called “twin dragon” was studied extensively by Knuth in his book [13].

Figure 1. The fundamental domain of a number system

Fundamental domains of number systems have been studied in various papers. Kátai and Kőrnyei [9] proved, that F is a compact set that tesselates the plane in the following way.

⋃_{g∈ℤ^{2}}(F + g) = ℝ^{2} where λ((F + g_{1}) ∩ (F + g_{2})) = 0 (g_{1}, g_{2} ∈ ℤ^{2}; g_{1} ≠ g_{2}). (1)

Furthermore, we want to mention, that the boundary of F has fractal dimension. Its Hausdorff and box counting dimension has been calculated by Gilbert [4], Ito [7], Müller-Thuswaldner-Tichy [16] and Thuswaldner [17]. In the present paper we are interested in topological properties of F. Before we give a survey on our results we shall define some basic objects. Let S be the set of all translates of F, that “touch” F, i.e.

S := {g ∈ ℤ^{2} \ (0, 0)^{T} | F ∩ (F + g) ≠ ∅}.

Then by (1) the boundary of F has the representation

∂F = ⋃_{g∈S}(F ∩ (F + g)).  (2)

Hence, the boundary of F is the set of all elements of F, that are contained in F + g for a certain g ≠ (0, 0)^{T}. Of course, ∂F may contain points, that belong to F and two other different translates of F. These points we call vertices of F. Thus the set of vertices of F is defined by

V := {z ∈ F | z ∈ (F + g_{1}) ∩ (F + g_{2}), g_{1}, g_{2} ∈ ℤ^{2}; g_{1} ≠ g_{2}, g_{1} ≠ 0, g_{2} ≠ 0}.

In Section 2 we study the set of vertices of F. It turns out, that, apart from one exceptional case, F has at least 6 vertices. In some cases we derive that V is an infinite or even uncountable set. In Section 3 we prove the connectedness of F and show that each element of F, which has a finite M-adic expansion, is an inner point of F.

2. Vertices of the Fundamental Domain F

In this section we give some results on the set of vertices V of F. For number systems emerging from Gaussian integers, similar results have been established with help of different methods in Gilbert [3]. We start with the definition of useful abbrevations. Let

g = M^{−H_{1}}d_{−H_{1}} + ⋯ + M^{H_{2}}d_{H_{2}}  (3)

be the M-adic representation of g. Note, that the digits d_{j} (−H_{1} ≤ j ≤ H_{2}) are of the shape d_{j} = (c_{j}, 0)^{T} ∈ Φ(N). Thus for the expansion (3) we will write

g = c_{H_{2}}c_{H_{2}−1} … c_{1}c_{0}.c_{−1} … c_{H_{1}}.

If the string c_{1} … c_{H} occurs j times in an M-adic representation, then we write [c_{1} … c_{H}]_{j}. If a representation is ultimately periodic, i.e. a string c_{1} … c_{H} occurs infinitely often, we write [c_{1} … c_{H}]_{∞}. First we show, that for A > 0 any fundamental domain F contains at least 6 vertices.

Theorem 2.1. Let (M, Φ(N)) be a number system in ℝ^{2}, which is induced by the base b of a canonical number system. Let p_{b}(x) = x^{2} + Ax + B with A > 0 be the minimal polynomial of b. Then the set of vertices V of the fundamental domain F of this number system contains the points

\begin{matrix}P_{1} = 0.[0(A − 1)(B − 1)]_{∞}, & P_{2} = 0.[(A − 1)(B − 1)0]_{∞}, \\ P_{3} = 0.[0(B − 1)(B − A)]_{∞}, & P_{4} = 0.[(B − 1)(B − A)0]_{∞}, \\ P_{5} = 0.[(B − 1)0(A − 1)]_{∞}, & P_{6} = 0.[(B − A)0(B − 1)]_{∞}. \end{matrix}

4 SHIGEKI AKIYAMA AND JÖRG M. THUSWALDNER

Depending on the cases A = 1, 1 < A < B and A = B, the points P_{j} (1 ≤ j ≤ 6) belong to the following translates F + w of F.

values of w for 1 < A < B  values of w for A = B
P_{1}  0, 1, 1A  0, 1, 1(A − 1)10
P_{2}  0, 1(A − 1), 1A(B − 1)  0, 1(A − 1), 1(A − 1)10(A − 1)
P_{3}  0, 1A, 1(A − 1)  0, 1(A − 1), 1(A − 1)10
P_{4}  0, 1A(B − 1), 1(A − 1)(B − A)  0, 1A(A − 1), 1(A − 1)0
P_{5}  0, 1(A − 1)(B − A + 1), 1  0, 1(A − 1)1, 1
P_{6} 0, 1(A − 1)(B − A), 1(A − 1)(B − A + 1)  0, 1(A − 1)1, 1(A − 1)0

The case A = 1 is very similar to the case 1 < A < B; just replace the representation 1(A − 1)(B − A + 1) by 11(B − 1)0 in the above table.

Remark 2.1. Note, that we have 0 < A ≤ B ≥ 2. Hence the digits of the 6 points indicated in Theorem 2.1 are all admissible.

Proof of the theorem. We will prove that each of the 6 points P_{1}, …, P_{6} is contained in three different translates of F, as indicated in the statement of the theorem. First we consider the point P_{1}. Write \overline{x} = −x. By using b^{2} + Ab + B = 0, we see that

0.1(A − 1)(B − A)\overline{B} = 0.1[(A − 1)(B − A)\overline{(B − 1)}]_{∞} = 0  (4)

are formal representations of zero. Adding the second representation for 0 given in (4) twice, we have

P_{1} = 0.[0(A − 1)(B − 1)]_{∞} + 1.[(A − 1)(B − A)\overline{(B − 1)}]_{∞}
= 1.[(A − 1)(B − 1)0]_{∞}
= 1.[(A − 1)(B − 1)0]_{∞} + 1(A − 1).[(B − A)\overline{(B − 1)}(A − 1)]_{∞}
= 1A.[(B − 1)0(A − 1)]_{∞}.

For A < B this yields

P_{1} ∈ F ∩ (F + 1) ∩ (F + 1A).

For A = B the last expansion 1A.[(B − 1)0(A − 1)]_{∞} is not admissible since A > B − 1. In order to settle this case we use the first representation of zero given in (4) to get 1A = 1B = 1B + 1(B − 1)0\overline{B} = 1(A − 1)10. As a result, we have

P_{1} ∈ F ∩ (F + 1) ∩ (F + 1(A − 1)10)

for A = B. Since P_{2} = MP_{1}, we get the desired results also for P_{2}. Now we treat

P_{3} = 0.[0(B − 1)(B − A)]_{∞}.

In the same way as before, we get, using both representations of zero in (4)

P_{3} = 0.[0(B − 1)(B − A)]_{∞} + 1A.B − 0.1[(A − 1)(B − A)\overline{(B − 1)}]_{∞}
= 1A.[(B − 1)(B − A)0]_{∞}
= 1A.[(B − 1)(B − A)0]_{∞} − 1.[(A − 1)(B − A)\overline{(B − 1)}]_{∞}
= 1(A − 1).[(B − A)0(B − 1)]_{∞},

which implies

P_{3} ∈ F ∩ (F + 1A) ∩ (F + 1(A − 1))

for A < B and

P_{3} ∈ F ∩ (F + 1(A − 1)10) ∩ (F + 1(A − 1))

for A = B. Since F permits an involution φ : x → ∑_{j≥1} M^{−j}(B −1, 0)^{T} −x, F is symmetric with respect to the center \frac{1}{2} ∑_{j≥1} M^{−j}(B − 1, 0)^{T}. For w ∈ ℤ^{2} this map sends each F + w to F − w. Thus we have

φ(F + 1) = F + 1A(B − 1),
φ(F + 1(A − 1)) = \begin{cases}F + 1(A − 1)(B − A + 1) & for A > 1, \\ F + 11(B − 1)0 & for A = 1, \end{cases}
φ(F + 1A) = F + 1(A − 1)(B − A),

for A < B and

φ(F + 1) = F + 1(A − 1)10(A − 1),
φ(F + 1(A − 1)) = F + 1(A − 1)1,
φ(F + 1(A − 1)10) = F + 1(A − 1)0,

for A = B. Furthermore, it is easy to see, that φ(P_{1}) = P_{4}, φ(P_{2}) = P_{5} and φ(P_{3}) = P_{6} Thus also P_{4}, P_{5} and P_{6} are vertices of F that are contained in the translates of F indicated in the statement of the theorem. □

In the case A = 0 it is easy to see that F is a square. It has exactly 4 vertices. These are the “usual” vertices of the square. Thus we only have to deal with the case A = −1. We will folmulate the corresponding result as a corollary.

Corollary 2.1. Let the same settings as in Theorem 2.1 be in force, but assume now, that A = −1. Then the following table gives 6 points P_{j} (1 ≤ j ≤ 6), that are contained in the set of vertices V of F. Furthermore, we give the translates F + w, to which P_{j} belongs.

P_{j}  translates w, for which P_{j} ∈ F + w
0.[0(B − 1)(B − 1)(B − 1)00]_{∞}  0, 10(B − 1), 10(B − 1)(B − 1)
0.[000(B − 1)(B − 1)(B − 1)]_{∞}  0, 1, 10
0.[00(B − 1)(B − 1)(B − 1)0]_{∞}  0, 10, 10(B − 1)
0.[(B − 1)000(B − 1)(B − 1)]_{∞}  0, 1, 10(B − 1)(B − 1)1
0.[(B − 1)(B − 1)000(B − 1)]_{∞} 0, 10(B − 1)(B − 1), 10(B − 1)(B − 1)1
0.[(B − 1)(B − 1)(B − 1)000]_{∞} 0, 10(B − 1)(B − 1)0, 10(B − 1)(B − 1)1

Proof. Let M_{1} = (\begin{matrix}0 & −B \\ 1 & −1 \end{matrix}) and M_{2} = (\begin{matrix}0 & −B \\ 1 & 1 \end{matrix}) be bases of number systems in ℝ^{2}and let

F_{1} and F_{2} be the fundamental domains corrresponding to M_{1} and M_{2}, respectively. We know the vertices of F_{1} from Theorem 2.1 and will construct the vertices of F_{2} from it. To this matter let M_{1} = G_{1}diag(b_{1}, b_{2})G_{1}^{−1}. It is easy to see, that then M_{2} = G_{2}diag(−b_{1}, −b_{2})G_{2}^{−1} with G_{2} = diag(−1, 1)G_{1}. Now suppose, that ∑_{k≥1} M_{1}^{−k}a_{j} ∈ F_{1} ∩ F_{1} + (v_{1}, v_{2})^{T} ∩ F_{1} +

6 SHIGEKI AKIYAMA AND JÖRG M. THUSWALDNER

(w_{1}, w_{2})^{T} with v_{1}, v_{2}, w_{1}, w_{2} ∈ ℤ is a vertex of F_{1}. Using the fact, that G_{1}^{−1}a_{k} = −G_{2}^{−1}a_{k} for a_{k} ∈ M and setting d = 0.[0(B − 1)]_{∞} we easily derive that

Q := ∑_{k≥1}(−1)^{k+1}M_{2}^{−k}a_{k} + d ∈ diag(−1, 1)(F_{1} ∩ F_{1} + (v_{1}, v_{2})^{T} ∩ F_{1} + (w_{1}, w_{2})^{T}) + d (5)

Observe, that by the selection of d, Q has an admissible M_{2}-adic representation with integer part zero. Thus Q ∈ F_{2}. Since any element of F_{2} can be constructed from elements of F_{1} in the same way we conclude, that F_{2} = diag(−1, 1)F_{1} + d. But with that (5) reads Q ∈ F_{2} ∩ F_{2} + (−v_{1}, v_{2})^{T} ∩ F_{2} + (−w_{1}, w_{2})^{T}. Thus Q is a vertex of F_{2}. The representations in the table above, can now easily be obtained from the results for A = 1 in Theorem 2.1. □

The following corollary is an immediate consequence of Theorem 2.1 and Corollary 2.1.

Corollary 2.2. For 1 < A < B we have

S ⊃ {1, 1A, 1(A − 1), 1A(B − 1), 1(A − 1)(B − A), 1(A − 1)(B − A + 1)},

for A = B

S ⊃ {1, 1(A − 1)10, 1(A − 1), 1(A − 1)10(A − 1), 1(A − 1)0, 1(A − 1)1},

while for A = 1

S ⊃ {1, 10, 10(B − 1), 10(B − 1)(B − 1), 10(B − 1)(B − 1)0, 10(B − 1)(B − 1)1}

holds.

Remark 2.2. Note, that “⊃” may be replaced by “=” in Corollary 2.2 if 2A < B + 3. This is shown for the Gaussian case in [16]. For arbitrary quadratic number fields this fact can be proved in a similar way.

Theorem 2.2. Let the same settings as in Theorem 2.1 be in force. If 2A = B + 3 then F has infinitely many vertices.

Proof. Set K = B − A + 1 = \frac{B−1}{2}. Then, using b^{2} + Ab + B = 0, we get (j ≥ 0)

\begin{aligned}0 & = & ∑_{k=2}^{∞}(−1)^{k} (M^{−k+2}(1, 0)^{T} + M^{−k+1}(A, 0)^{T} + M^{−k}(B, 0)^{T} ) \\ & & = 1.(A − 1)[K\overline{K}]_{∞}. \end{aligned}  (6)

Here we set \overline{x} = −x, as before. We will show, that the points

Q_{j} = 1A.[(B − 1)0(A − 1)]_{2j}(B − 1)0[K]_{∞} (j ∈ ℕ)  (7)

are vertices of F. Therefore we need the representation (6). With help of this representation we define the following representations of zero.

N_{1} := 1(A − 1).[K\overline{K}]_{∞} = 0,
N_{2} := 1.(A − 1)[K\overline{K}]_{∞} = 0,
X_{j} := 0.[0]_{j}1AB = 0 (j ≥ 0).

In the sequel we write kX_{j} (k ∈ ℤ) if we want to multiply each digit of the representation X_{j} by k. Furthermore, addition and subtraction of representations is always meant digit-wise. After these definitions we define the following, more complicated representations of zero.

Z_{1}(j) := N_{1} + ∑_{k=1}^{j}(X_{6k−1} − 2X_{6k−2} + 2X_{6k−3} − X_{6k−4}) + (1.AB) − 2(1A.B)
= \overline{1}\overline{A}.[(\overline{B − 1})(A − 1)(B − A)]_{2j}(\overline{B − 1})(A − 1)[K\overline{K}]_{∞},
Z_{2}(j) := N_{2} + ∑_{k=1}^{j}(−X_{6k−3} + 2X_{6k−4} − 2X_{6k−5} + X_{6k−6}) − (1A.B)
= \overline{1}(\overline{A − 1}).(\overline{B − A})[(B − 1)(\overline{A − 1})(\overline{B − A})]_{2j−1}(B − 1)(\overline{A − 1})\overline{K}[K\overline{K}]_{∞}.

Finally, we observe, that for j ∈ ℕ

Q_{j} = Q_{j} + Z_{1}(j)
= 0.[0(A − 1)(B − 1)]_{2j}0(A − 1)[(B − 1)0]_{∞}
= Q_{j} + Z_{2}(j)
= 1.(A − 1)[(B − 1)0(A − 1)]_{2j−1}(B − 1)0KK[0(B − 1)]_{∞},

and this implies Q_{j} ∈ V. It remains to show, that the elements Q_{j}, j ≥ 1, are pairwise different. This follows from the following observation. Select k ∈ ℕ arbitrary and let j_{1}, j_{2} ≤ k. Suppose, that Q_{j_{1}} and Q_{j_{2}} are represented by the representation (7) for j = j_{1} and j = j_{2}, respectively. Then Q_{j_{1}} = Q_{j_{2}} if and only if M^{6k+2}Q_{j_{1}} = M^{6k+2}Q_{j_{2}}. For k ≥ max(j_{1}, j_{2}), M^{6k+2}Q_{j_{1}} and M^{6k+2}Q_{j_{2}} have the same digit string [0(B − 1)]_{∞} after the comma. Hence, they can only be equal, if their integer parts are equal. But since (M, Φ(N)) is a number system, this can only be the case, if the digit strings of their integer parts are the same. This implies j_{1} = j_{2}. So we have proved, that the points Q_{j} are pairwise different for j ≤ k. Since k can be selected arbitrary, the result follows. Thus we found infinitely many different vertices of F. □

Theorem 2.3. Let the same settings as in Theorem 2.1 be in force. If 2A > B + 3 then F has uncountably many vertices.

Proof. Set K = B − A + 1 and ξ = ⌊(B − 1)/2⌋. As ξ + K, ξ − K ∈ N, by using (6), we see that

0.[ξ]_{∞} = 1(A − 1).[(ξ + K)(ξ − K)]_{∞}
= 1(A − 1)K.[(ξ − K)(ξ + K)]_{∞}.

Thus 0.[ξ]_{∞} is a vertex of F. Fix an integer k, such that all eigenvalues of M^{k} are greater than 2 (such an integer exists, since the eigenvalues of M are all greater than 1). This implies, that the representations 0.c_{1}[0]_{k}c_{2}[0]_{k}c_{3}[0]_{k}c_{4} …, c_{j} ∈ {0, 1} (j ≥ 1) represent pairwise different elements of ℝ^{2} for different {0, 1} sequences {c_{j}}_{j≥1}. Because ξ + K < B − 1, each of the uncountably many representations

0.[ξ]_{∞} + 0.c_{1}[0]_{k}c_{2}[0]_{k}c_{3}[0]_{k}c_{4} … (c_{j} ∈ {0, 1}, j ≥ 1)

corresponds to a vertex of F. Since they are pairwise different, the theorem is proved. □

3. Connectedness and Inner Points of the Fundamental Domain F

In this section we will show, that the fundamental domain F is arcwise connected. To establish this result, we will apply a general theorem due to Hata (cf. [5, 6]) which assures arcwise connectedness for a large class of sets. The second result of this section is devoted to the structure of the inner points of F. In particular, we prove, that each point with finite M-adic representation is an inner point of F. In this section we will use the notation

F_{k} := {z | z = ∑_{j=1}^{k} M^{−j}a_{j}, a_{j} ∈ Φ(N)} (k ∈ ℕ).

We start with the connectedness result.

Theorem 3.1. Let (M, Φ(N)) be a number system in ℝ^{2}, which is induced by the base b of a canonical number system in a quadratic number field. Then the fundamental domain F of (M, Φ(N)) is arcwise connected.

Proof. It is an easy consequence of the definition of F, that

F = ⋃_{g∈Φ(N)} M^{−1}(F + g).  (8)

Furthermore, Theorem 2.1 implies that F ∩ (F + (1, 0)^{T}) ≠ ∅. Thus the sets contained in the union of (8) form a chain in the sense that (F + g) ∩ (F + (g + (1, 0)^{T})) ≠ ∅ for g ∈ Φ(N) \ (B − 1, 0)^{T}. Thus F fulfills the conditions being necessary for the application of a theorem of Hata, namely [5, Theorem 4.6]. This theorem yields the arcwise connectedness of F. □

Now we prove the result on the inner points of F. Note, that the existence of inner points is an immediate consequence of [9, Theorem 1].

Theorem 3.2. Let (M, Φ(N)) be a number system in ℝ^{2}, which is induced by the base b of a canonical number system in a quadratic number field. Then for each k ∈ ℕ we have

F_{k} ⊂ int(F).

Proof. First we will show, that 0 is an inner point of F. Suppose, that 0 is contained in the boundary of F. Then by (2) there exists a representation of zero of the shape

0 = c_{H_{1}}c_{H_{1}−1} … c_{1}c_{0}.c_{−1}c_{−2} ….  (9)

This representation implies 0 ∈ F + c_{H_{1}}c_{H_{1}−1} … c_{1}c_{0}. If we multiply (9) by M^{j} for j ∈ ℕ arbitrary, we conclude, that 0 ∈ F + c_{H_{1}}c_{H_{1}−1} … c_{1}c_{0}c_{−1} … c_{−j} for each j ∈ ℕ. Hence, 0 is contained in infinitely many different translates of F. But since F is a compact set this is a contradiction to (1). Thus 0 ∈ int(F).

Now fix k ∈ ℕ and g ∈ F_{k}. Then 0 ∈ int(F) implies, that g ∈ int(M^{−k}F + g). The result now follows from the representation

F = ⋃_{g∈F_{k}}(M^{−k}F + g).

□

There is a direct alternative proof of this theorem by using the methods of [1] and [2]. In these papers a similar result for the tiling generated by Pisot number systems is shown.

References

[1] S. Akiyama. Self affine tiling and pisot numeration system. In K. Győry and S. Kanemitsu, editors, Number Theory and its Applications. Kluwer Academic Publishers. to appear.

[2] S. Akiyama and T. Sadahiro. A self-similar tiling generated by the minimal pisot number. Acta Math. Info. Univ. Ostraviensis, 6:9–26, 1998.

[3] W. J. Gilbert. Complex numbers with three radix representations. Can. J. Math., 34:1335–1348, 1982.

[4] W. J. Gilbert. Complex bases and fractal similarity. Ann. sc. math. Quebec, 11(1):65–77, 1987.

[5] M. Hata. On the structure of self-similar sets. Japan J. Appl. Math, 2:381–414, 1985.

[6] M. Hata. Topological aspects of self-similar sets and singular functions. In J. Bélair and S. Dubuc, editors, Fractal Geometry and Analysis, pages 255–276, Netherlands, 1991. Kluwer Academic Publishers.

[7] S. Ito. On the fractal curves induced from the complex radix expansion. Tokyo J. Math., 12(2):299–320, 1989.

[8] I. Kátai. Number systems and fractal geometry. preprint.

[9] I. Kátai and I. Kőrnyei. On number systems in algebraic number fields. Publ. Math. Debrecen, 41(3– 4):289–294, 1992.

[10] I. Kátai and B. Kovács. Kanonische Zahlensysteme in der Theorie der Quadratischen Zahlen. Acta Sci. Math. (Szeged), 42:99–107, 1980.

[11] I. Kátai and B. Kovács. Canonical number systems in imaginary quadratic fields. Acta Math. Hungar., 37:159–164, 1981.

[12] I. Kátai and J. Szabó. Canonical number systems for complex integers. Acta Sci. Math. (Szeged), 37:255–260, 1975.

[13] D. E. Knuth. The Art of Computer Programming, Vol 2: Seminumerical Algorithms. Addison Wesley, London, 3rd edition, 1998.

[14] B. Kovács. Canonical number systems in algebraic number fields. Acta Math. Hungar., 37:405–407, 1981.

[15] B. Kovács and A. Pethő. Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math. (Szeged), 55:286–299, 1991.

[16] W. Müller, J. M. Thuswaldner, and R. F. Tichy. Fractal properties of number systems. preprint.

[17] J. M. Thuswaldner. Fractal dimension of sets induced by bases of imaginary quadratic fields. Math. Slovaca, 48(4):365–371, 1998.

Shigeki Akiyama  Jörg M. Thuswaldner
Department of Mathematics Department of Mathematics and Statistics
Faculty of Science  Montanuniversität Leoben
Niigata University  Franz-Josef-Str. 18
NIIGATA  LEOBEN
JAPAN  AUSTRIA