Uniqueness and topological properties of number representation
Let b be a complex number with |b| > 1 and let D be a finite subset of the complex plane C such that 0 ∊ D and card D ≥ 2. A number z is representable by the system (D, b) if z = Σajbj , where aj ∊ D. We denote by F the set of numbers which are representable by (D, b) with M = −1. The set W consi...
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nasplib_isofts_kiev_ua-123456789-1246222025-02-09T11:37:47Z Uniqueness and topological properties of number representation Dovgoshey, O. Martio, O. Ryazanov, V. Vuorinen, M. Let b be a complex number with |b| > 1 and let D be a finite subset of the complex plane C such that 0 ∊ D and card D ≥ 2. A number z is representable by the system (D, b) if z = Σajbj , where aj ∊ D. We denote by F the set of numbers which are representable by (D, b) with M = −1. The set W consists of numbers that are (D, b) representable with aj = 0 for all negative j. Let F1 be a set of numbers in F that can be uniquely represented by (D, b). It is shown that: The set of all extreme points of F is a subset of F1. If 0 ∊ F1, then W is discrete and closed. If b ∊ {z : |z| > 1}\D′, where D′ is a finite or countable set associated with D and W is discrete and closed, then 0 ∊ F1. For a real number system (D, b), F is homeomorphic to the Cantor set C iff F\F1 is nowhere dense subset of R. 2004 Article Uniqueness and topological properties of number representation / O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen // Український математичний вісник. — 2004. — Т. 1, № 3. — С. 331-348. — Бібліогр.: 12 назв. — англ. 1810-3200 2000 MSC. 11A67. https://nasplib.isofts.kiev.ua/handle/123456789/124622 en Український математичний вісник application/pdf Інститут прикладної математики і механіки НАН України |
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Let b be a complex number with |b| > 1 and let D be a finite subset of the complex plane C such that 0 ∊ D and card D ≥ 2. A number z is representable by the system (D, b) if z = Σajbj , where aj ∊ D. We denote by F the set of numbers which are representable by (D, b) with M = −1. The set W consists of numbers that are (D, b) representable with aj = 0 for all negative j. Let F1 be a set of numbers in F that can be uniquely represented by (D, b). It is shown that: The set of all extreme points of F is a subset of F1. If 0 ∊ F1, then W is discrete and closed. If b ∊ {z : |z| > 1}\D′, where D′ is a finite or countable set associated with D and W is discrete and closed, then 0 ∊ F1. For a real number system (D, b), F is homeomorphic to the Cantor set C iff F\F1 is nowhere dense subset of R. |
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Dovgoshey, O. Martio, O. Ryazanov, V. Vuorinen, M. |
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Dovgoshey, O. Martio, O. Ryazanov, V. Vuorinen, M. Uniqueness and topological properties of number representation Український математичний вісник |
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Dovgoshey, O. Martio, O. Ryazanov, V. Vuorinen, M. |
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Uniqueness and topological properties of number representation |
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Uniqueness and topological properties of number representation |
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Uniqueness and topological properties of number representation |
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Uniqueness and topological properties of number representation |
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Uniqueness and topological properties of number representation |
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uniqueness and topological properties of number representation |
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Інститут прикладної математики і механіки НАН України |
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Uniqueness and topological properties of number representation / O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen // Український математичний вісник. — 2004. — Т. 1, № 3. — С. 331-348. — Бібліогр.: 12 назв. — англ. |
| series |
Український математичний вісник |
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AT dovgosheyo uniquenessandtopologicalpropertiesofnumberrepresentation AT martioo uniquenessandtopologicalpropertiesofnumberrepresentation AT ryazanovv uniquenessandtopologicalpropertiesofnumberrepresentation AT vuorinenm uniquenessandtopologicalpropertiesofnumberrepresentation |
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2025-11-25T21:46:19Z |
| last_indexed |
2025-11-25T21:46:19Z |
| _version_ |
1849800455942569984 |
| fulltext |
Український математичний вiсник
Том 1 (2004), № 3, 331 – 348
Uniqueness and topological properties of
number representation
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen
Abstract. Let b be a complex number with |b| > 1 and let D be a
finite subset of the complex plane C such that 0 ∈ D and card D ≥ 2. A
number z is representable by the system (D, b) if z =
M∑
j=−∞
ajb
j , where
aj ∈ D. We denote by F the set of numbers which are representable
by (D, b) with M = −1. The set W consists of numbers that are (D, b)
representable with aj = 0 for all negative j. Let F1 be a set of numbers in
F that can be uniquely represented by (D, b). It is shown that: The set
of all extreme points of F is a subset of F1. If 0 ∈ F1, then W is discrete
and closed. If b ∈ {z : |z| > 1}\D′, where D′ is a finite or countable set
associated with D and W is discrete and closed, then 0 ∈ F1. For a real
number system (D, b), F is homeomorphic to the Cantor set C iff F\F1
is nowhere dense subset of R.
2000 MSC. 11A67.
Key words and phrases. Representations of numbers, Cantor sets.
1. Introduction
Suppose we have a finite setD of complex numbers, 0 ∈ D, cardD ≥ 2
and a number b ∈ C, |b| > 1. We denote by F the set of ”fractions” for
the system (D, b) and by W the corresponding set of integers:
f ∈ F ⇐⇒ f =
−1∑
j=−∞
ajb
j , (1.1)
w ∈W ⇐⇒ w =
M∑
j=0
kjb
j , (1.2)
Received 11.02.2004
The first and third authors thank for the support Department of Mathematics of Uni-
versity of Helsinki
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
332 Uniqueness and topological properties...
where aj and kj belong to D.
A ”general” number q is representable by the system (D, b) iff
q =
M∑
j=−∞
ajb
j , aj ∈ D, (1.3)
i.e., g = w + f , w ∈ W , f ∈ F . We shall write G for the set of all
representable numbers, by definitions (1.1), (1.2) and (1.3)
G = F +W.
The definitions of F and W and various examples of real and complex
number systems can be found in [4]. See also [3], [5], [6] for an information
about representability of complex numbers by the special complex sys-
tems. Topological properties of real number representations were studied
in more general situations, in [2], [9], [11].
The purpose of this work is the investigation of similarities between
the uniqueness of the representations by the system (D, b) and topological
properties of F , W and G.
To avoid ambiguities we recall the following definition.
Definition 1.1. Let f be an element of the set F . The element f has a
unique representation in the form (1.1) iff for any two series
−1∑
j=−∞
k
(1)
j bj
and
−1∑
j=−∞
k
(2)
j bj, where all k
(1)
j and k
(2)
j belong to D:
f =
−1∑
j=−∞
k
(1)
j bj =
−1∑
j=−∞
k
(2)
j bj
=⇒ (k
(1)
j = k
(2)
j )
for each negative integer j.
Let F1 = F1(D, b) denote the set of numbers that can be uniquely ex-
pressed as (1.1) and let F2 = F\F1 be the corresponding complementary
subset of F . Similarly, we introduce sets W1, G1, W2 and G2: w ∈ W1
iff w has a unique representation (1.2); g ∈ G1 iff g has a unique repre-
sentation (1.3); W2 = W\W1 and G2 = G\G1.
2. Statements of results
It should be noted that some numbers have a single representation
in the one form but the same numbers may fail to have the single rep-
resentation in another form. The first three propositions illuminate this
phenomen.
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 333
Proposition 2.1. Let (D, b) be a number system. Then the following
three properties are equivalent:
F2 6= ∅; (2.1)
G2 6= ∅; (2.2)
(F − F ) ∩ ((D −D)\{0}) 6= ∅, (2.3)
where F − F = {x− y : x ∈ F, y ∈ F} and
(D −D)\{0} = {x− y : x ∈ D, y ∈ D, x 6= y}.
Proposition 2.2. Let (D, b) be a number system. Then the following
two properties are equivalent:
F1 ∩G2 6= ∅; (2.4)
(F1 − F ) ∩ (D\{0}) 6= ∅, (2.5)
where F1 − F = {x− y : x ∈ F1, y ∈ F} and
D\{0} = {x : x ∈ D, x 6= 0}.
Example 2.1 Let (D, b) be the usual binary system: D = {0, 1}, b = 2.
Then we have 0 ∈ F , 1 ∈ F1 ∩G2 and 1 = 1 − 0.
Let B2F = B2F (D) and B2W = B2W (D) be the subsets of {z ∈ C :
|z| > 1} defined by the next relations:
(b ∈ B2F ) ⇐⇒ (F2(D, b) 6= ∅), (2.6)
(b ∈ B2W ) ⇐⇒ (W2(D, b) 6= ∅). (2.7)
Proposition 2.3. Let D be a finite set of complex numbers, card D ≥ 2,
0 ∈ D. Then:
2.3.1. B2W is at most countable and nonempty;
2.3.2. B2F ⊇ [−2,−1) ∪ (1, 2].
Example 2.2 Let b = 3 and D = {0, 2}. Then F is the Cantor ternary
set C. In this case, it is known that F2 = ∅. Consequently, by Proposi-
tion 2.1, G2 = ∅ and from W2 ⊆ G2 follows W2 = ∅.
If (D, b) is a number system, then the convex hull of F will be denoted
by F̂ . The set of all extreme points of F̂ will be denoted by Ext F̂ . The
following theorem shows that there is no number system with F1 = ∅.
334 Uniqueness and topological properties...
Theorem 2.1. Let (D, b) be a number system. Then Ext F̂ is subset of
F1. In symbols,
ExtF̂ ⊆ F1. (2.8)
Corollary 2.1. Let (D, b) be a complex (real) number system. Then F1
is a nonempty Gδ subset of C (of R) and F2 is Fσ subset of C (of R).
Example 2.3 Let (D, b) be the standard decimal system:D={0, 1, . . . , 9}
and b = 10. Then we have that: F = [0, 1], ExtF = {0, 1}, 0 ∈ F1 ∩G1
and 1 ∈ F1 ∩G2.
Remark 2.1. The set of all extreme points of an arbitrary closed convex
plane set is closed [1, Exercise 11.9.8]. Since F is compact, Ext F̂ is a
compact subset of F .
Theorem 2.2. Let (D, b) be a number system. If 0 ∈ G1, then W is
closed and discrete in C.
Theorem 2.3. Let D be a finite set of complex numbers, card D ≥ 2,
0 ∈ D. Suppose b ∈ {z : |z| > 1}\B2W . If W is closed and discrete in C,
then 0 ∈ G1(D, b).
Remark 2.2. By Proposition 2.3 the set B2W is at most countable and
hence Theorem 2.12 is an ”almost converse” of Theorem 2.2.
Example 2.4 Let b = 10 and D = {1, 1,−9}. Then b ∈ B2W , zero is not
in G1(D, b), but W is closed and discrete.
Theorem 2.4. Let (D, b) be a number system. Then the following three
statements are equivalent:
2.4.1. F is homeomorphic to the Cantor ternary set C;
2.4.2. The small inductive dimension of G is zero. In symbols,
ind G = 0;
2.4.3. ind F 2 ≤ 0.
Corollary 2.2. Let (D, b) be a real number system. Then F is homeo-
morphic to C iff F2 is a nowhere dense subset of R.
Remark 2.3. By the definition of small inductive dimension we have
idnF 2 = −1 iff F 2 = ∅.
The following two propositions define more precisely some aspects of
Theorem 2.4 and Corollary 2.2.
Proposition 2.4. Let (D, b) be a number system. If cardD = 2 and
b ∈ (1,+∞), then F is homeomorphic to C iff F2 is empty.
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 335
Proposition 2.5. If n ∈ N, b ∈ C and |b| > n ≥ 3, then there exists a
finite set D ⊆ C such that card D = n, 0 ∈ D, F (D, b) is homeomorphic
to C and F2(D, b) 6= ∅.
Our final theorem gives some survey of topological properties of num-
ber representation by systems with F2 = ∅.
Theorem 2.5. Let (D, b) be a number system. If F2(D, b) = ∅, then:
2.5.1. F is compact, perfect, zero-dimensional, that is homeomorphic
to the Cantor set C;
2.5.2. W is a closed, discrete and unbounded subset of C;
2.5.3. G is closed, perfect and zero-dimensional subset of C.
Remark 2.4. For an arbitrary (D, b), we have the following: F is com-
pact and perfect; W is unbounded; and if W is a closed subset of C, then
G is closed, too.
Vector generalizations. Many our propositions and theorems re-
main valid when one passes from a number system to the following many-
dimensional construction: D is a finite set in Rn, including zero, and B
is n × n nonsingular matrix with a norm ‖B‖ > 1. It should also be
observed that Theorem 2.1 remains valid for a positional vector system
whose definition similar to Definition 2.1 from the Petkovs̆ek’s work [9].
3. Proofs
3.1. Proof of Proposition 2.1
The trivial inclusion
F2 ⊆ F ∩G2 (3.1.1)
shows that the implication (2.1) ⇒ (2.2) is correct. Let x be an element
of the set G2. By the definition of G2 there are two sequences {aj} and
{a′j} for which
x =
M∑
j=−∞
ajb
j =
M∑
j=−∞
a′jb
j (3.1.2)
holds with
M∑
j=−∞
|aj − a′j | 6= 0. Let j0 be the greatest subscript with
|aj0 − a′j0 | 6= 0. Then using (3.1.2) we obtain
aj0 +
j0−1∑
j=−∞
ajb
j−j0 = a′j0 +
j0−1∑
j=−∞
a′jb
j−j0 , (3.1.3)
336 Uniqueness and topological properties...
where aj0 6= a′j0 . The last equality is equivalent to (2.3). So we have only
to establish implication (2.3) ⇒ (2.1). Suppose that (3.1.3) holds with
aj0 6= a′j0 . Then taking the number t as
t =
j0∑
j=−∞
ajb
j−j0−1 =
j0∑
j=−∞
a′jb
j−j0−1,
we have that t ∈ F2. Hence we get F2 6= ∅.
3.2. Proof of Proposition 2.2
Suppose d is an element of (F1 − F ) ∩ (D\{0}). Then d ∈ D, d 6= 0
and d = t1 − t, with t1 ∈ F1 and t ∈ F . Hence t1 = d+ t ∈ F1 ∩G2, and
we have (2.5) ⇒ (2.4). Now suppose f is an element of G2 ∩ F1. Since
f ∈ F1 we have a unique representation
f =
−1∑
j=−∞
ajb
j , (3.2.1)
where each aj ∈ D. Let F = F(f) be the family of all representations of
f which are different from (3.2.1). Since f ∈ G2, we have that F 6= ∅.
Let (s) be an element of F
(s) =
f =
Ms∑
j=−∞
a
(s)
j bj
, (3.2.2)
and let j0 = j0(s) be the greatest subscript for which a
(s)
j0
6= 0. Since
f ∈ F1, we have j0(s) ≥ 0. Now to prove the implication (2.4) ⇒ (2.5),
it suffices to justify the equality
min {j0(s) : (s) ∈ F(f), f ∈ F1 ∩G2} = 0. (3.2.3)
Consider any number f0 ∈ F1 ∩G2 with a representation (s) ∈ F(f0)
such that
(s) =
f0 =
M∑
j=−∞
ajb
j
aM 6= 0,
M = min {j0(s) : (s) ∈ F(f), f ∈ F1 ∩G2}.
In order to check that (3.2.3) holds it is sufficient to show that
(M > 0) ⇒ (b−1f0 ∈ G2 ∩ F1).
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 337
It is clear that f0 ∈ G2 implies b−1f0 ∈ G2. Suppose f0 ∈ G2 ∩ F1,
M > 0 and b−1f0 ∈ F2. By the last supposition we can find two different
representations
−1∑
j=−∞
a
(1)
j bj =
−1∑
j=−∞
a
(2)
j bj = b−1f0. (3.2.4)
If |a(1)
−1| + |a(2)
−1| = 0 holds, then it follows from (3.2.4) that
f0 =
−2∑
j=−∞
a
(1)
j bj+1 =
−2∑
j=−∞
a
(2)
j bj+1.
This contradicts to f0 ∈ F1. Consequently, |a(1)
−1| + |a(2)
−1| 6= 0, and we
have
f0 = a
(1)
−1 +
−2∑
j=−∞
a
(1)
j bj+1 = a
(2)
−1 +
−2∑
j=−∞
a
(2)
j bj+1,
contrary to the assumption M > 0.
3.3. Proof of Proposition 2.3
Lemma 3.1. Let D be a finite set of complex numbers with card D ≥ 2
and 0 ∈ D. Then a complex number b belongs to B2W iff |b| > 1 and
there is a polynomial p(z) =
n∑
i=0
aiz
i such that p(b) = 0, n ≥ 1, an 6= 0
and ai ∈ (D −D) for i = 0, 1, . . . , n.
Lemma 3.2. The polynomial p(z) = z3 − z + 1 has a real root z0 with
|z0| > 1.
Lemma 3.3. Let D1 ⊆ D be two finite sets of complex numbers, and let
0 ∈ D1, card D1 ≥ 2. Then
F2(D1, b) ⊆ F2(D, b),
for each b with |b| > 1.
Lemma 3.4. Let (D, b) be a number system and let z be a nonzero
complex number. Then
Fi(zD, b) = zFi(D, b).
for i = 1, 2.
The simple proofs of these lemmas are omitted.
338 Uniqueness and topological properties...
Lemma 3.5. Let b be a real number with |b| > 1 and let D = {0, 1},
then F2(D, b) is nonempty if and only if
b ∈ [−2,−1) ∪ (1, 2].
Proof. It follows from Proposition 2.1 that F2(D, b) is nonempty iff there
exists a sequence {aj}−1
−∞ whose elements belong to the set {−1, 0, 1} and
1 =
−1∑
j=−∞
ajb
j . (3.3.1)
Hence in the case D = {0, 1} we have the equivalence
(F2(D, b) = ∅) ≡ (F2(D,−b) = ∅). (3.3.2)
Consequently, we shall restrict ourselve, to the case b > 1. If b > 2, then
−1∑
j=−∞
|ajbj | < 1 and equality (3.3.1) cannot holds. It therefore remains
to verify that 1 is a distance between two points of F (D, b) for b ∈ (1, 2].
It follows directly from the early Randolph’s result [10]
Theorem 3.1 (Randolph). Let {an}∞ be a sequence with an > 0,
a1 ≥ a2 ≥ . . ., and
∑∞
n=1 an = 1. For a fixed {an}∞n=1, let S be the set
of all sums of the form
∑
εnan where εn is equal 1 or 0. Then the set
S − S fills the unit interval [0, 1] iff
an ≤
∞∑
k=n+1
ak.
We can now easily prove Proposition 2.3.
Lemma 3.1 implies that B2W is at most countable, and from Lemmas
3.1 and 3.2 it follows that B2W is nonempty. By Lemmas 3.5 and 3.4 we
have B2F ⊇ [−2,−1)∪ (1, 2] for each two-point set D, and using Lemma
3.3 we have (2.3.2) for the case card D > 2.
3.4. Proof of Theorem 2.1
Let z0 be an extreme point of F̂ . Since Ext F̂ ⊆ ∂F̂ , there is a straight
line l0 which contains z0, and one of its closed half-plane includes F̂ [8,
Theorem 3.2]. This is a so-called straight line of support of a convex
set F̂ .
For the sake of simplicity, suppose l0 and real axis are mutually per-
pendicular,
l0 = {z ∈ C : Re z = Re z0}.
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 339
This we can always do by choosing the suitable Θ ∈ [0, 2π) and pass-
ing on to the set eiΘD = {eiΘd1, . . . , e
iΘdk} from the "old" set D =
{d1, . . . , dk}. Passing to eiΘD we obtain eiΘF, eiΘF1, e
iΘF̂ and eiΘExt F̂
from F, F1, F̂ and Ext F̂ . We can assume, without loss of generality,
that
Re z ≤ Re z0 (3.4.1)
for all z ∈ F̂ .
Consider first the case where
l0 ∩ F̂ = {z0}. (3.4.2)
For any negative integer j, define Dj by the rule:
(a ∈ Dj) ⇐⇒ (a ∈ D and Re(abj) = max
d∈D
Re(dbj)). (3.4.3)
Since D is finite and nonempty, we have Dj 6= ∅ for each negative integer
j. Let t0 be the number with a representation
t0 =
−1∑
j=−∞
ajb
j ,
where aj ∈ Dj for j = −1,−2, . . . .
We claim that t0 = z0. It is obvious that t0 is an element of F . From
the definition of extreme point we have Ext F̂ ⊆ F [8, Theorem 4.2].
Hence z0 ∈ F and, by (3.4.3) Re z0 ≤ Re t0. The reverse inequality
follows from (3.4.1). Consequently, Re z0 = Re t0. From the last equality
and (3.4.2) we have t0 = z0.
The equality z0 = t0 implies that Dj has the unique element for each
negative integer j. Really, given any negative integer j0, we fix elements
a
(1)
j0
and a
(2)
j0
of the set Dj0 , then for any sequence {aj} such that aj ∈ Dj
we have
a
(1)
j0
bj0 +
−1∑
j=−∞
j 6=j0
ajb
j = z0 = a
(2)
j0
bj0 +
−1∑
j=−∞
j 6=j0
ajb
j .
Hence a
(1)
j0
= a
(2)
j0
and Dj0 is an one-point set.
We can now easily show that z0 ∈ F1. If there are two representations
z0 =
−1∑
j=−∞
cjb
j , cj ∈ D
and
z0 =
−1∑
j=−∞
ajb
j , aj ∈ Dj ,
340 Uniqueness and topological properties...
then by (3.4.3) the inequality
Re(cjb
j) ≤ Re(ajb
j) (3.4.4)
holds for each negative integer j but
−1∑
j=−∞
Re(cjb
j) =
−1∑
j=−∞
Re(ajb
j). (3.4.5)
The relations (3.4.4) and (3.4.5) imply the equality
Re(cjb
j) = Re(ajb
j)
for each negative integer j. Since Dj is an one-point set, we have cj = aj
for all j.
Consider now the case where
∃ z1 ∈ l0 ∩ F̂ : z1 6= z0.
We can restrict ourselves to the situation of the inequality Im z1 < Im z0.
From the last inequality it follows that
∀ z ∈ F̂ ∩ l0 : Im z ≤ Im z0. (3.4.6)
(In the opposite case, z0 is an interior point of the interval [z1, z2] where
z2 is some point of F̂ . This contradicts to the inclusion z0 ∈ Ext F̂ .)
For any negative integer j, define Do
j by the rule:
(a ∈ Do
j ) ⇐⇒ (a ∈ Dj and Im(abj) = max
d∈Dj
Im(dbj)) (3.4.7)
where Dj was defined by (3.4.3). We claim that Do
j is an one-point set.
Let j be a negative integer, and let a1, a2 be elements of Do
j . Then we
have:
Re(a1b
j) = Re(a2b
j) = max
d∈D
Re(dbj),
Im(a2b
j) = Im(a2b
j) = max
d∈Dj
Im(dbj).
Hence a2b
j = a1b
j holds. Since b 6= 0, it follows that a1 = a2.
Let us denote by aj the unique element of Do
j . Consider an arbitrary
representation of z0,
z0 =
−1∑
j=−∞
cjb
j ,
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 341
where cj ∈ D for each j. Now to prove that z0 ∈ F1 it suffices to
demonstrate the equality cj = aj for each negative integer j.
Set t0 :=
−1∑
j=−∞
ajb
j where aj ∈ Do
j . As it has been proved above,
(3.4.1) and (3.4.3) imply Re z0 = Re t0, and hence t0 ∈ F̂ ∩ l0. It follows
from the equality Re z0 = Re t0 that
cj ∈ Dj (3.4.9)
for each negative integer j. The relation t0 ∈ F̂ ∩ l0 and (3.4.6) imply the
inequality
(3.4.10)
−1∑
j=−∞
Im(ajb
j) ≤
−1∑
j=−∞
Im(cjb
j).
By formulal (3.4.7) and (3.4.9) we have
Im(cjb
j) ≤ Im(ajb
j), j = −1,−2, . . . .
From this and (3.4.10) it follows that Im(cjb
j) = Im(ajb
j), and hence
cj ∈ Do
j for each negative integer j. Since Do
j = {aj}, the equality
aj = cj hold for all negative integer j.
3.5. Proof of Corollary 2.1
We may assume without loss of generality that (D, b) is a complex
number system. Since a convex hull of a compact subset of Rn is compact
[8, Theorem 2.6] and F is a compact subset of C [4, Proposition 2.2.23],
it follows that F̂ is compact, and by the Krein-Milman theorem we have
that Ext F̂ 6= ∅ [8, Corollary of Theorem 4.2]. The last inequality and
(2.8) imply that F1 6= ∅.
We turn to the proof that F2 is a Fσ. Let us denote by Dω the
product of a countable collection of copies of the discrete space D =
{d1, . . . , dk}. As usual, we assume that Dω has a product (Tychonoff)
topology. The classic Tychonoff theorem implies that Dω is a compact
space. All elements of Dω can be regarded as sequences {aj}−1
j=−∞ with
aj ∈ D for each negative integer j. Define a map Φ : Dω → F by the
rule: if a = {aj}−1
j=−∞ ∈ Dω, then
Φ(a) =
−1∑
j=−∞
ajb
j . (3.5.1)
It is easy to see that Φ is continuous and onto.
342 Uniqueness and topological properties...
Let j0 be a negative integer and let d ∈ D. Then we set
Πj0
d := {a ∈ Dω : a = (a−1, a−2, . . .), aj0 = d}.
All Πj
d are closed subsets of the compact Dω, and hence all Πj
d are com-
pact.
From the definition 1.1 it follows that
F2 =
−1⋃
j=−∞
k−1⋃
i=1
k⋃
l=i+1
(Φ(Πj
di
) ∩ Φ(Πj
dl
)) (3.5.2)
where di and dl are elements of the set D = {d1, . . . , dk}. Since a con-
tinuous image of a compact set is compact, Φ(Πj
d) is closed for each Πj
d.
Hence, by formula (3.5.2) F2 is a Fσ.
The definition of F1 implies that
F1 = (C\F2) ∩ F. (3.5.3)
Since for a metric spaces each closed set is Gδ, it follows that F is Gδ.
The complement of an Fσ is Gδ, hence C\F2 is Gδ. Therefore, by (3.5.3)
F1 is Gδ.
3.6. Proof of Theorem 2.2
Suppose there is either a point t0 ∈ W\W or a point t0 ∈ W ′ where
W is the closure of W and W ′ is the set of all accumulation points of W .
In the both cases, we can find a sequence {zn}, n ∈ N, such that:
lim
n→∞
zn = t0; ∀n ∈ N : zn ∈W ; (3.6.1)
∀n,m ∈ N : (n 6= m) ⇒ (zn 6= zm).
For each zn there exists a representation
zn =
Qn∑
j=0
a
(n)
j bj
where a
(n)
j ∈ D and Qn ≥ 0. Using conditions (3.6.1), we can find a
subsequence {znk
}, znk
= a
(nk)
Qnk
bQnk + . . .+ a
(nk)
0 b0, of the sequence {zn}
for which
∀nk ∈ N : a
(nk)
Qnk
6= 0 and Qnk+1
> Qnk
. (3.6.2)
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 343
We may assume without loss of generality that {znk
} and {zn} coincide.
Conditions (3.6.1) and (3.6.2) imply that
lim
n→∞
(a
(n)
Qn
+ a
(n)
Qn−1
b−1 + . . .+ a
(n)
0 b−Qn) = lim
n→∞
t0
bQn
= 0. (3.6.3)
Since D is finite and for each n ∈ N : 0 6= a
(n)
Qn
∈ D, there exists a constant
infinite subsequence {a(nk)
Qnk
} of {a(n)
Qn
} such that
a
(nk)
Qnk
= d (3.6.4)
with d ∈ D, d 6= 0. Now we can, once again, take nk = n. Put
∆n := a
(n)
Qn−1b
−1 + . . .+ a
(n)
0 b−Qn .
The equalities (3.6.3) and (3.6.4) show that
lim
n→∞
∆n = −d.
Since ∆n ∈ F and F is compact, we have −d ∈ F . From the definition
of F it follows that the nonzero number −d has a representation
−d =
−1∑
j=−∞
ajb
j .
Hence
0 = d− d = d+
−1∑
j=−∞
ajb
j ,
that is 0 ∈ G2.
3.7. Proof of Theorem 2.3
Suppose 0 is not in G1(D, b) but W is closed and discrete in C. It
suffices to show that these assumptions imply b ∈ B2W (D). By the
supposition 0 ∈ G2(D, b), and hence we can find a representation
0 =
Q∑
j=−∞
ajb
j (3.7.1)
with
Q∑
j=−∞
|aj | 6= 0 and aj ∈ D for each j. If there is some k < Q such
that aj = 0 for all j < k, then
0 =
Q∑
j=−∞
ajb
j−k =
Q−k∑
j=0
aj+kb
j .
344 Uniqueness and topological properties...
It follows, in this case, that b ∈ B2W (D) (see Lemma 3.1). Hence we can
restrict ourselves to the case when
∀ j < Q ∃ k < j : |ak| 6= 0.
Let n be a positive integer. If n ≥ Q, then by (3.7.1)
0 = bn
Q∑
j=−∞
ajb
j = wn + fn
where
wn :=
Q∑
j=−n
ajb
j+n and fn :=
−n−1∑
j=−∞
ajb
j+n.
Since fn ∈ F for each n and F is compact, there is a convergent subse-
quence {fni
} of the sequence {fn}. The equality wn+fn = 0 implies that
{wni
} is convergent, too. Let w be the limit of {wni
}. By the assumptions
W is discrete and closed. Consequently, for some i0, we have
w = wni0
= wni0+1 = wni0+2 = . . . .
This implies that w ∈W2. Thus b ∈ B2W (D).
3.8. Proof of Theorem 2.4
Lemma 3.6. Let (D, b) be a number system. Then the set F is a compact
perfect set.
Proof. It is known that F is compact. It remains to show that every
f ∈ F is an accumulation point of F . By the definition of F we have
f =
−1∑
j=−∞
ajb
j
with aj ∈ D. Fix two different elements d1 and d2 of the set D. Let i be
a negative integer. Setting
a
(i)
j :=
aj if j 6= i,
d1 if j = i and aj 6= d1,
d2 if j = i and aj = d1
and
fi :=
−1∑
j=−∞
a
(i)
j b
j ,
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 345
we obtain the sequence {fi}−1
i=−∞ such that
lim
i→−∞
fi = f,
and fi ∈ F , fi 6= f , for each i.
Lemma 3.7. Let (D, b) be a number system. If F2 6= ∅, then F2 is a
dense subset of the set F .
Proof. Let f0 be an element of F2. It is easy verified that bjf0 ∈ F2 for
each negative integer j. Let f be an element of F . By the definition of
F we have
f =
−1∑
j=−∞
aj(f)bj
where aj(f) ∈ D. For each negative integer k, define fk by the formula:
fk := bk−1f0 +
−1∑
j=k
aj(f)bj .
Then fk ∈ F2 for each k and
lim
k→−∞
fk = f.
Now the proof of Theorem 2.4 follows from the properties of zero-
dimensional sets and Lemmas 3.6, 3.7, see below.
(2.4.1) ⇒ (2.4.2) If F is homeomorphic to the Cantor set C, then
F is closed and zero-dimensional. An union of a countable family of zero-
dimensional closed sets in a separable metric space is zero-dimensional
[4, Corollary 3.2.9]. Since W is countable and
G =
⋃
w∈W
(w + F ),
the set G is zero-dimensional.
(2.4.2) ⇒ (2.4.3) Suppose that ind G = 0. Since F 2 ⊆ F ⊆ G, we
have ind F 2 ≤ ind G = 0 [4, Theorem 3.1.7].
(2.4.3) ⇒ (2.4.1) Consider first the case where ind F 2 = 0. In
this case, F2 is a nonvoid set and by Lemma 3.7 we have F = F 2. Using
Lemma 3.6, we have that F is a compact, perfect zero-dimensional subset
of the complex plane C. Hence F is homeomorphic to the Cantor ternary
346 Uniqueness and topological properties...
set [12, Theorem 29.7 and Corollary 30.4]. Now, suppose that indF 2 =
−1. By the definition of the small inductive dimension we have F2 = ∅,
i. e., each element of the set F has a unique representation (1.1). Under
this condition the map Φ : Dω → F (see formula 3.5.1) is one-to-one,
continuous and onto. Hence Φ is a homeomorphism [12, Theorem 17.14].
Since every two totally disconnected, perfect, compact metrizable spaces
are homeomrphic, it follows that the Cantor set C is homeorphic to Dω.
[12, Theorem 30.3 and Corollary 30.4]. Consequently, C is homeomorphic
to F .
3.9. Proof Corollary 2.2
Let us denote by Int F 2 the set of all interior points of the set F 2.
We must show that Int F 2 = ∅ iff ind F 2 ≤ 0. This follows directly from
the well-known
Theorem 3.2. [7, Theorem IV.3] Let Rn be the Eucliden n–dimensional
space, and let A ⊆ Rn. Then ind A = n iff Int A 6= ∅.
3.10. Proof of Proposition 2.4
We may assume without loss of generality that D = {0, 1}, (see
Lemma 3.4).
Lemma 3.8. [10] If b1 ≥ b2 ≥ b3 ≥ . . . , bn . . . > 0,
∞∑
n=1
bn = s < ∞
and bn ≤
∞∑
i=n+1
bi, then corresponding to any number z, 0 ≤ z ≤ s, there
exists a sequence {εn} each element of which is either 0 or 1, such that
z =
∞∑
n=1
εnbn.
Suppose b is a point in the interval (1, 2]. Then from Randolph’s
Lemma 3.8 it follows that [0, 1] ⊆ F , and so F cannot be homeomophic
to the Cantor set C. Observe also that by Lemma 3.5 we have F2 6= ∅
for b ∈ (1, 2]. If b ∈ (2,∞), then by Lemma 3.5 F2 = ∅ and Theorem 2.4
shows that F is homeomorphic to C.
3.11. Proof of Proposition 2.5
For an arbitrary number system (D, b) with D = {d1, . . . , dk} we
construct the corresponding iterated function system {f1, . . . , fk} where
fj : C → C, fj(z) = b−1z + dj , j = 1, . . . , k dj ∈ D.
O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen 347
It is easy to see that F = F (D, b) is the invariant set for this iterated
system, that is
F =
k⋃
j=1
fj(F ),
and similarity dimension of F is
s(F ) = lg(k)/ lg(b)
(see, for example, [4, Chapter 4]). Since ind F ≤ s(F ) [4, Theorem 6.2.10
and Theorem 6.3.8], it follows from Theorem 2.4 that if cardD < |b|, then
F is homeomorphic to the Cantor set C.
Hence, if n ∈ N, n ≥ 3, b ∈ C, |b| > n, {0, 1, b−1} ⊆ D ⊆ C,
card D = n, then F (D, b) is homeomorphic to C and by Proposition 2.1
F2(D, b) 6= ∅.
3.12. Proof of Theorem 2.5
Suppose W is closed. It is enough to show that G is closed.
Let g be an accumulation point of G. Then there is a sequence
{gn}∞n=1 such that g = lim
n→∞
gn and gn ∈ G for each n. By the defi-
nition of G we have gn = fn +wn where fn ∈ F and wn ∈W . Since F is
compact, there is a convergent subsequence {fnk
} of the sequence {fn}.
Set
f := lim
k→∞
fnk
.
Then we have
g − f = lim
k→∞
wnk
,
and since W is closed, it follows that g − f ∈W . Hence
g = (g − f) + f ∈W + F = G.
References
[1] Marcel Berger, Géométrie. Cedic, Paris, 1977; Nathan, Paris 1977.
[2] F. S. Cater, Real numbers with redundant representations // Real Anal. Ex-
change, 17 (1991–92), 282–290.
[3] Chandler Davis and Donald J. Knuth, Number representations and dragon
curves // Part I: J. Recreational Math., 3 (1970), 66–81; Part II: J. Recre-
ational Math., 3 (1970), 133–149.
[4] Gerald A. Edgar, Measure, Topology and Fractal Geometry. Springer–Verlag,
New York — Berlin — Heidelberg, 1992.
[5] W. J. Gilbert, Fractal geometry derived from complex bases // Math. Intelli-
gencer. 4 (1982), 78–86.
348 Uniqueness and topological properties...
[6] W. J. Gilbert, The fractal dimension of sets derived from complex bases //
Canad. Math. Bull. 29 (1986), 495–500.
[7] Witold Hurewicz and Henry Wallman, Dimension Theory. Princeton University
Press, Princeton, 1948.
[8] K. Leichtweiss, Konvexe Menger. VEB Deutscher Verlag der Wissenschaften,
Berlin, 1980.
[9] M. Petkovs̆ek, Ambiquous number are dense // Amer. Math. Monthly, 97 1990,
408–411.
[10] J. F. Randolph, Some properties of sets of the Cantor type // J.London Math.
Soc. 16 (1941), 38–42.
[11] M. Starbird and T. Starbird, Required redundancy in the representation of re-
als // Proc. Amer. Math. Soc. 114 (1992), 796–774.
[12] Stephen Willard, General Topology. Addison — Wesley Publishing Company,
Reading, Massachusetts–Menlo Park, California — London — Don Mills, On-
tario — Sydney, 1970.
Contact information
O. Dovgoshey and
V. Ryazanov
Institute of Applied Mathematics
and Mechanics, NAS of Ukraine,
74 Roze Luxemburg str.,
Donetsk, 83114
Ukraine
E-Mail: dovgoshey@iamm.ac.donetsk.ua,
ryaz@iamm.ac.donetsk.ua
O. Martio and
V. Vuorinen
Department of Mathematics,
P.O. Box 4 (Yliopistonkatu 5), FIN–00014
University of Helsinki,
Finland
E-Mail: martio@cc.helsinki.fi,
vuorinen@csc.fi
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