Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits
Dedicated to V. S. Korolyuk on occasion of his 80-th birthday Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic e...
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Інститут математики НАН України
2005
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irk-123456789-1658262020-02-17T01:27:47Z Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits Pratsiovytyi, M.V. Torbin, H.M. Статті Dedicated to V. S. Korolyuk on occasion of his 80-th birthday Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proven that the set Tₛ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ( Tₛ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented. Детально вивчаються властивості множини Tₛ „особливо ненормальних чисел" одиничного інтервалу (тобто множини чисел x, для яких немає асимптотичної частоти деяких цифр в s-адичному зображенні, а деякі цифри мають асимптотичні частоти). Доведено, що множина Tₛ є нехтуваною в топологічному сенсі (першої категорії Бера) та загальною в сенсі фрактальної геометрії (Tₛ є суперфрактальною множиною, розмірність Хаусдорфа-Безиковича якої дорівнює одиниці). Наведено топологічну і фрактальну класифікацію множин дійсних чисел через аналіз асимптотичної частоти їх s-адичних зображень. 2005 Article Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits / M.V. Pratsiovytyi, H.M. Torbin // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1163–1170. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165826 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Pratsiovytyi, M.V. Torbin, H.M. Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits Український математичний журнал |
| description |
Dedicated to V. S. Korolyuk on occasion of his 80-th birthday
Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proven that the set Tₛ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ( Tₛ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented. |
| format |
Article |
| author |
Pratsiovytyi, M.V. Torbin, H.M. |
| author_facet |
Pratsiovytyi, M.V. Torbin, H.M. |
| author_sort |
Pratsiovytyi, M.V. |
| title |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_short |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_full |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_fullStr |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_full_unstemmed |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_sort |
singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
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Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165826 |
| citation_txt |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits / M.V. Pratsiovytyi, H.M. Torbin // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1163–1170. — Бібліогр.: 8 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT pratsiovytyimv singularprobabilitydistributionsandfractalpropertiesofsetsofrealnumbersdefinedbytheasymptoticfrequenciesoftheirsadicdigits AT torbinhm singularprobabilitydistributionsandfractalpropertiesofsetsofrealnumbersdefinedbytheasymptoticfrequenciesoftheirsadicdigits |
| first_indexed |
2025-07-14T20:05:05Z |
| last_indexed |
2025-07-14T20:05:05Z |
| _version_ |
1837654087467794432 |
| fulltext |
UDC 519.21
S. Albeverio (Inst. Angew. Math.; Univ. Bonn, Germany),
M. Pratsiovytyi (Nat. Ped. Univ., Kyiv),
G. Torbin (Nat. Ped. Univ., Kyiv; Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
SINGULAR PROBABILITY DISTRIBUTIONS
AND FRACTAL PROPERTIES OF SETS OF REAL NUMBERS
DEFINED BY THE ASYMPTOTIC FREQUENCIES
OF THEIR s-ADIC DIGITS
SYNHULQRNI JMOVIRNISNI ROZPODILY
TA FRAKTAL\NI VLASTYVOSTI MNOÛYN
DIJSNYX ÇYSEL, WO ZADANI
ASYMPTOTYÇNOG ÇASTOTOG }X s-ADYÇNYX CYFR
Dedicated to V. S. Korolyuk
on occasion of his 80-th birthday
Properties of the set Ts of “particularly nonnormal numbers” of the unit interval are studied in details (Ts
consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating
s-adic expansion of x, and some do not). It is proven that the set Ts is residual in the topological sense (i.e.,
it is of the first Baire category) and it is generic in the sense of fractal geometry ( Ts is a superfractal set, i.e., its
Hausdorff – Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers
via analysis of asymptotic frequencies of digits in their s-adic expansions is presented.
Detal\no vyvçagt\sq vlastyvosti mnoΩyny Ts ,,osoblyvo nenormal\nyx çysel” odynyçnoho inter-
valu (tobto mnoΩyny çysel x, dlq qkyx nema[ asymptotyçno] çastoty deqkyx cyfr v s-adyçnomu
zobraΩenni, a deqki cyfry magt\ asymptotyçni çastoty). Dovedeno, wo mnoΩyna Ts [ nextuvanog
v topolohiçnomu sensi (perßo] katehori] Bera) ta zahal\nog v sensi fraktal\no] heometri] (Ts [ su-
perfraktal\nog mnoΩynog, rozmirnist\ Xausdorfa – Bezykovyça qko] dorivng[ odynyci). Navedeno
topolohiçnu i fraktal\nu klasyfikacig mnoΩyn dijsnyx çysel çerez analiz asymptotyçno] çastoty
]x s-adyçnyx zobraΩen\.
1. Introduction. Let us consider the classical s-adic expansion of x ∈ [0, 1] :
x =
∞∑
n=1
s−nαn(x) = ∆sα1(x)α2(x) . . . αk(x) . . . , αk(x) ∈ A = {0, 1, . . . , (s−1)},
and let Ni(x, k) be the number of digits “i” among the first k digits of the s-adic
expansion of x , i ∈ A. If the limit νi(x) = lim
k→∞
Ni(x, k)
k
exists, then the number
νi(x) is said to be the frequency of the digit “i” (or the asymptotic frequency of “i”) in
the s-adic expansion of x .
A property of an element x ∈ M is usually said to be “normal” if “almost all” ele-
ments of M have this property. There exist many mathematical notions (e.g., cardinality,
measure, Hausdorff – Besicovitch dimension, Baire category) allowing us to interpret the
words “almost all” in a rigorous mathematical sense. “Normal” properties of real num-
bers are deeply connected with the asymptotic frequencies of their digits in some systems
of representation.
The set
Ns =
{
x
∣∣∣∣(∀i ∈ A) lim
k→∞
Ni(x, k)
k
=
1
s
}
c© S. ALBEVERIO, M. PRATSIOVYTYI, G. TORBIN, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9 1163
1164 S. ALBEVERIO, M. PRATSIOVYTYI, G. TORBIN
is said to be the set of s-normal numbers (or the set of real numbers which are normal
with respect to the base s). It is well known (E. Borel, 1909), that the sets Ns and the set
N∗ =
∞⋂
s=2
Ns are of full Lebesgue measure (i.e., they have Lebesgue measure 1).
The unit interval [0, 1] can be decomposed in the following way:
[0, 1] = Es
⋃
Ds,
where
Es = {x|νi(x) exists ∀i ∈ A},
Ds =
{
x
∣∣∣∣∃i ∈ A, lim
k→∞
Ni(x, k)
k
does not exist
}
.
The set Ds is said to be the set of nonnormal real numbers. Each of the subsets Es and
Ds can be decomposed in the following natural way.
The set
Ws =
{
x
∣∣∣∣(∀i ∈ A) : νi(x) exists and (∃j ∈ A) : νj(x) �=
1
s
}
is said to be the set of quasinormal numbers. It is evident that
Es = Ws
⋃
Ns, Ws
⋂
Ns = ∅.
The set
Ls =
{
x
∣∣∣∣(∀i ∈ A) lim
k→∞
Ni(x, k)
k
does not exist
}
is said to be the set of essentially nonnormal numbers.
The set
Ts =
{
x
∣∣∣∣(∃i ∈ A) : lim
k→∞
Ni(x, k)
k
does not exist, and
(∃j ∈ A) : lim
k→∞
Nj(x, k)
k
exists
}
is said to be the set of particularly nonnormal numbers.
It is evident that
Ds = Ls
⋃
Ts, Ls
⋂
Ts = ∅.
The sets Ns, Ws, Ts, Ls are everywhere dense sets, because the frequencies νi(x)
do not depend on any finite number of s-adic symbols of x . It is also not hard to prove
that these sets have the cardinality of the continuum.
The main purpose of the paper is to fill in completely the following table, which
reflects the metric, topological and fractal properties of the corresponding sets:
Lebesgue measure Hausdorff dimension Baire category
Ns
Ws
Ls
Ts
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SINGULAR PROBABILITY DISTRIBUTIONS AND FRACTAL PROPERTIES OF SETS . . . 1165
Let ν = (ν0, ν1, . . . , νs−1) be a stochastic vector and let
Ws[ν] =
{
x : x = ∆sα1(x)α2(x) . . . αk(x) . . . ,
lim
k→∞
N∗
i (x, k)
k
= νi ∀i ∈ A
}
.
The well known Besicovitch – Eggleston’s theorem (see, e.g., [1, 2]) gives the following
formulae for the determination of the Hausdorff – Besicovitch dimension α0 (Ws[ν]) of
the set Ws[ν] :
α0 (Ws[ν]) =
∑s−1
k=0
νi log νi
− log s
.
From the latter formulae it easily follows that the set Ws of all quasinormal numbers
is a superfractal set, i.e., Ws is a set of zero Lebesgue measure with full Hausdorff –
Besicovitch dimension ( α0 (Ws) = 1).
Properties of subsets of the set of nonnormal numbers have been intensively studied
during recent years (see, e.g., [3 – 6] and references therein). Some interesting subsets of
Ds were studied in [4] by using the techniques and results from the theory of multifractal
divergence points. In [3] it has been proven that the set Ds is superfractal.
In the paper [7] of the authors it has been proven that the set Ls of essentially non-
normal numbers is also superfractal and it is of the second Baire category. Moreover,
it has been proven that the set Ls contains an everywhere dense Gδ -set. So, the sets
Ns,Ws, Ts are of the first Baire category. From these results it follows that essentially
nonnormal numbers are generic in the topological sense as well as in the sense of fractal
geometry; nevertheless, the set Ls is small from the point of view of Lebesgue measure.
The main goal of the present paper is the investigation of fractal properties of the set
Ts of particularly nonnormal numbers. To this end we apply a probabilistic approach
for the calculation of the Hausdorff dimension of subsets. More precisely, we apply the
results of fine fractal analysis of singular continuous probability distributions.
The first step of the fractal analysis of a singular continuous measure ν is the in-
vestigation of metric, topological and fractal properties of the corresponding topological
support Sν (i.e., the minimal closed set supporting the measure). These are good charac-
teristics only for the class of uniform Cantor-type singular measures. But, in general, they
are only ”external characteristics”, because there exist essentially different singular con-
tinuous measures concentrating on the common topological support. The main idea of the
paper [7] consisted in the construction of singular continuous measures whose topological
supports coincide with some subsets of the set of essentially nonnormal numbers.
The second step of the fractal analysis of a singular continuous measure ν is the
determination of the Hausdorff dimension α0(ν) (and the local Hausdorff dimension )
of the measure, i.e., roughly speaking, finding the Hausdorff dimension of the minimal (in
the fractal dimension sense) supports (which are not necessarily closed) of the measure.
This problem is much more complicated than the previous one (see, e.g., [8]), especially
in the case of essentially superfractal measures.
In Section 2 we prove that for all s ≥ 3 the set Ts is of full Hausdorff dimension.
To prove the main result we construct a sequence of singular continuous measures µp
such that the corresponding minimal dimensional supports consist of only particularly
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1166 S. ALBEVERIO, M. PRATSIOVYTYI, G. TORBIN
nonnormal numbers, and apply the results of [8] to perform a fine fractal analysis of these
supports.
2. Fractal properties of the set of particularly nonnormal numbers. Let us study
the sets Ts of particularly nonnormal numbers which were defined in Section 1. It is easy
to see that the set T2 is empty, because from the existence of the asymptotic frequency
νi(x) for some i ∈ {0, 1} the existence of another asymptotic frequency follows.
Theorem 1. For any positive integer s ≥ 3 the set Ts of particularly nonnormal
real numbers is superfractal, i.e., the Hausdorff – Besicovitch dimension of the set Ts
equals 1.
Proof. To prove the theorem we shall construct a superfractal set G ⊂ Ts .
In the sequel we usually shall not use the indices s in the notation of the corre-
sponding subsets, since s will be an arbitrary fixed natural number greater than 2. Let
us consider the classical s-adic expansion of x ∈ [0, 1] : x =
∑∞
n=1
s−nαn(x) =
= ∆sα1(x)α2(x) . . . αk(x) . . . . If x is an s-adic rational number, then we shall use the
representation without the period s− 1.
For a given p ∈ N and for any x ∈ [0, 1) we define the following mapping ϕp :
ϕp(x) = ϕp (∆sα1(x)α2(x) . . . αk(x) . . .) =
= ∆s
s−1︷ ︸︸ ︷
00 . . . 0
s−1︷ ︸︸ ︷
11 . . . 1 . . .
s−1︷ ︸︸ ︷
(s− 2)(s− 2) . . . (s− 2)(s− 1)α1(x)α2(x) . . . αs2p(x)
2(s−1)︷ ︸︸ ︷
00 . . . 0
2(s−1)︷ ︸︸ ︷
11 . . . 1 . . .
2(s−1)︷ ︸︸ ︷
(s− 2)(s− 2) . . . (s− 2)
(s− 1)(s− 1)αs2p+1(x)αs2p+2(x) . . . αs2p+2s2p(x) . . .
. . .
2k−1(s−1)︷ ︸︸ ︷
00 . . . 0 11 . . . 1 . . .
2k−1(s−1)︷ ︸︸ ︷
(s− 2)(s− 2) . . . (s− 2)
2k−1︷ ︸︸ ︷
(s− 1)(s− 1) . . . (s− 1)
α(2k−1−1)s2p+1(x) . . . α(2k−1)s2p(x) . . . .
Let us explain the construction of ϕp . First of all we divide the s-adic expansion
of x into groups in the following way: the k-th group consists of the sequence
(α(2k−1−1)s2p+1(x) . . . α(2k−1)s2p(x)), k ∈ N . The s-adic expansion of y = ϕp(x)
is constructed from the s-adic expansion of x via inserting (before the k-th group) the
following series of fixed symbols (0 . . . 01 . . . 1 . . . (s− 2) . . . (s− 2)(s− 1) . . . (s− 1)),
where each symbol i (0 ≤ i ≤ s− 2) occurs 2k−1(s− 1) times, but the symbol s− 1
occurs 2k−1 times.
Let Mp = ϕp([0, 1)) = {y : y = ϕp(x), x ∈ [0, 1)}.
For a given p ∈ N and for any y ∈ Mp we define the mapping ψp(y) in the
following way: if
y = ϕp(x) = ∆s
s−1︷ ︸︸ ︷
00 . . . 0 . . .
s−1︷ ︸︸ ︷
(s− 2) . . . (s− 2)(s− 1)α1(x)α2(x) . . . αs2p(x)
2(s−1)︷ ︸︸ ︷
0 . . . 0 . . .
2(s−1)︷ ︸︸ ︷
(s− 2) . . . (s− 2)(s− 1)(s− 1)
αs2p+1(x)αs2p+2(x) . . . αs2p+2s2p(x) . . . ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
SINGULAR PROBABILITY DISTRIBUTIONS AND FRACTAL PROPERTIES OF SETS . . . 1167
then
z = ψp(y) = ∆s
s−1︷ ︸︸ ︷
0 . . . 0(s− 1) . . .
s−1︷ ︸︸ ︷
(s− 2) . . . (s− 2) . . .
. . . (s− 1)(s− 1) (01 . . . (s− 2))α1(x)α2(x) . . . αs2p(x)
(s−1)︷ ︸︸ ︷
00 . . . 0(s− 1)
(s−1)︷ ︸︸ ︷
00 . . . 0(s− 1) . . .
(s−1)︷ ︸︸ ︷
(s− 2)(s− 2) . . . (s− 2)
(s− 1)
(s−1)︷ ︸︸ ︷
(s− 2)(s− 2) . . . (s− 2)(s− 1)
(s− 1) (01 . . . (s− 2)) (s− 1) (01 . . . (s− 2))αs2p+1(x)αs2p+2(x) . . .
. . . αs2p+2s2p(x) . . . , x ∈ [0, 1),
i.e., the s-adic expansion of z = ψp(y) can be obtained from the s-adic expansion of
y = ϕ(x) by using the following algorithm:
1) after any fixed symbol (s−1) we insert the following series of symbols: (01 . . . (s−
− 2));
2) after any subseries consisting of (s − 1) fixed symbols i (0 ≤ i ≤ s − 2) we
insert the symbol s− 1.
Let fp = ψp(ϕp) and let
Sp = fp([0, 1)) = {z : z = fp(x), x ∈ [0, 1)} = {z : z = ψp(y), y ∈Mp},
Gp = fp([0, 1)) = {z : z = fp(x), x ∈ Ns}.
The following two lemmas will describe some properties of the constructed sets Gp .
Lemma 1. For any z =
∑∞
n=1
s−nαn(z) ∈ Gp the lim
n→∞
Ni(z, n)
n
does not exist
for any i ∈ {0, 1, . . . , s− 2}, and lim
n→∞
Ns−1(z, n)
n
=
1
s
.
Proof. The set Gp has the following structure:
Gp =
=
{
z:z=∆s
s−1︷︸︸︷
0...0 (s−1)...
s−1︷ ︸︸ ︷
(s−2)...(s−2)(s−1)(s−1)(01...(s−2))α1(x)α2(x)...αs2p(x)︸ ︷︷ ︸
first group
0...0(s−1)0...0(s−1)...(s−2)...(s−2)(s−1)(s−2)...(s−2)(s−1)(s−1)(01...︸ ︷︷ ︸
second
...(s−2))(s−1)(01...(s−2))αs2p+1(x)αs2p+2(x)...αs2p+2s2p(x)︸ ︷︷ ︸
group
. . . , x∈Ns
}
.
From x ∈ Ns it follows that the symbol s−1 has the asymptotic frequency
1
s
in the se-
quence {αk(x)} and the equality lim
n→∞
Ns−1(z, n)
n
=
1
s
follows from the construction
of the set Gp .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1168 S. ALBEVERIO, M. PRATSIOVYTYI, G. TORBIN
Let lk be the number of the position at which the above k-th group of symbols ended,
i.e., lk = s2(p+ 1)(2k − 1).
Let m′
k(i) be the number of the position at which the k-th series of the fixed symbols
i and (s− 1) (0 ≤ i ≤ s− 2) ended, i.e., m′
k+1(i) = s2(p+ 1)(2k − 1) + s(i+ 1)2k.
Let m′′
k(i) be the number of the position at which the k-th series of the fixed symbols
i (0 ≤ i ≤ s− 2) started, i.e., m′′
k+1(i) = s2(p+ 1)(2k − 1) + si2k + 1 .
If z ∈ Gp , then there are s(2k+1 − 1) + dk) symbols i (0 ≤ i ≤ s − 2) among
the first m′
k+1(i) symbols of the s-adic expansion of z , where dk is the quantity of
the symbol i among the first (2k − 1)s2p s-adic symbols αi(x) in the expansion of
x = f−1
p (z). Since x is an s-normal number, we have dk = (2k − 1)sp+ o(2k).
So,
lim
n→∞
Ni(z,m′
k+1(i))
m′
k+1(i)
=
= lim
n→∞
(2k+1 − 1)s+ (2k − 1)sp+ s−1o(2k)
s2(p+ 1)(2k − 1) + s(i+ 1)2k
=
p+ 2
s(p+ 1) + i+ 1
.
If z ∈ Gp , then there are s(2k −1)+dk symbols i (0 ≤ i ≤ s−2) among the first
m′′
k+1(i) − 1 symbols of the s-adic expansion of z .
So,
lim
n→∞
Ni(z,m′′
k+1(i) − 1)
m′′
k+1(i) − 1
= lim
n→∞
(2k − 1)s+ (2k − 1)sp+ s−1o(2k)
s2(p+ 1)(2k − 1) + si2k
=
=
p+ 1
s(p+ 1) + i
<
p+ 2
s(p+ 1) + i+ 1
.
Therefore, for any z ∈ Gp and for any i ∈ {0, 1, . . . , s − 2} the limit lim
n→∞
Ni(z, n)
n
does not exist.
The lemma is proved.
The following Corollary is immediate, using the definitions of Gp, Ts and Lemma 1:
Corollary 1. Gp ⊂ Ts ∀p ∈ N.
Lemma 2. The Hausdorff – Besicovitch dimension of the set Gp is equal to
p
p+ 2
.
Proof. Let Bp(i) be the subset of N with the following property: ∀k ∈ N, k ∈
∈ Bp(i) if and only if αk(fp(x)) = i for any x ∈ [0, 1) , i.e., Bp(i) consists of the
numbers of positions with the fixed symbols i in the s-adic expansion of any z ∈ Sp .
Let Bp =
s−1⋃
i=0
Bp(i) , and let Cp = N \Bp .
Let us consider the following random variable ξ(p) with independent s-adic digits:
ξ(p) =
∞∑
k=1
s−k ξ
(p)
k ,
where ξ(p)
k are independent random variables with the following distributions: if k ∈
∈ Bp(i) , then ξ(p)
k takes the value i with probability 1. If k ∈ Cp , then ξ(p)
k takes the
values 0, 1, . . . , (s− 1) with probabilities
1
s
,
1
s
, . . . ,
1
s
.
It is evident that the set Sp is the topological support of the distribution of the ran-
dom variable ξ(p) . Actually, the corresponding probability measure µp = Pξ(p) is the
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SINGULAR PROBABILITY DISTRIBUTIONS AND FRACTAL PROPERTIES OF SETS . . . 1169
image of Lebesgue measure on [0, 1) under the mapping fp = ψp(ϕp) , i.e., ∀E ⊂ B :
µp(E) = µp(E
⋂
Sp) = λ(f−1
p (E
⋂
Sp)) .
A. Firstly we prove that α0(Gp) ≤
p
p+ 2
. Since Gp ⊂ Sp, it is sufficient to show
that α0(Sp) ≤ p
p+ 2
. To this end we consider the sequence
{
B
(k)
i
}
(k ∈ N, i ∈
∈ {1, 2, . . . , ss2p(2k−1−1)}) of special coverings of the set Sp by s-adic closed intervals
of the rank mk = lk − 2k−1s2p = s2(p + 1)(2k − 1) − 2k−1s2p . For any k ∈ N
the covering
{
B
(k)
i
}
consists of the ss
2p(2k−1−1) closed s-adic intervals of mk-th rank
with length εk = s−(s2(p+1)(2k−1)−2k−1s2p).
The α-volume of the covering
{
B
(k)
i
}
is equal to
lαεk
(Sp) = ss
2p(2k−1−1)s−α(s2(p+1)(2k−1)−2k−1s2p) = s(p−α(p+2))2k−1s2
sα(p+1)−p.
For the Hausdorff premeasure hα
εk
we have hα
εk
(Sp) ≤ lαεk
(Sp) for any k ∈ N . So,
for the Hausdorff measure Hα we have Hα(Sp) ≤ lim
k→∞
lαεk
(Sp) = 0 if α >
p
p+ 2
.
Hence, α0(Sp) ≤
p
p+ 2
.
B. Secondly we prove that α0(Gp) ≥ p
p+ 2
. To this end we shall analyze the
internal fractal properties of the singular continuous measure µp .
For any probability measure ν one can introduce the notion of the Hausdorff dimen-
sion of the measure in the following way:
α0(ν) = inf
E∈N(ν)
{α0(E), E ∈ B},
where N(ν) is the class of all “possible supports” of the measure ν , i.e.,
N(ν) = {E : E ∈ B, ν(E) = 1}.
An explicit formula for the determination of the Hausdorff dimension of the measures
with independent Q∗-symbols has been found in [8]. Applying this formula to our case(
qik =
1
s
, ∀k ∈ N, ∀i ∈ {0, 1, . . . , s− 1}
)
, we have
α0(µp) = lim
n→∞
Hn
nlns
,
where Hn =
∑n
j=1
hj , and hj are the entropies of the random variables ξ(p)
j : hj =
= −
∑s−1
i=0
pij ln pij .
If j ∈ Bp , then hj = 0 . If j ∈ Cp , then hj = ln s .
So,
α0(µp) = lim
n→∞
Hn
nlns
= lim
k→∞
Hmk
mk lns
=
= lim
k→∞
s2p(2k−1 − 1) ln s
(s2(p+ 1)(2k − 1) − ps22k−1) ln s
=
p
p+ 2
.
The above defined set Gp = fp(Ns) is a support of the measure µp , because µp =
= λ(f−1
p ) and the Lebesgue measure of the set Ns of s-normal numbers of the unit
interval is equal to 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1170 S. ALBEVERIO, M. PRATSIOVYTYI, G. TORBIN
Since Gp ∈ N(µp) and α0(µp) =
p
p+ 2
, we get α0(µp) ≥ p
p+ 2
, which proves
Lemma 2.
Corollary 2. The set Gp is the minimal dimensional support of the measure µp ,
i.e., α0(Gp) ≤ α0(E) for any other support E of the measure µp.
Finally, let us consider the set G =
∞⋃
p=1
Gp. From Lemma 1 it follows that G ⊂ Ts .
From Lemma 2 and from the countable stability of the Hausdorff dimension it follows
that α0(G) = sup
p
α0(Gp) = 1. So, α0(Ts) = 1, which proves Theorem 1.
Summarizing the results of Sections 1 and 2, we have for s > 2 :
Lebesgue measure Hausdorff dimension Baire category
Ns 1 1 first
Ws 0 1 first
Ts 0 1 first
Ls 0 1 second
For the case s = 2 we have a corresponding result, but the Hausdorff dimension of
the set Ts is equal to 0, because the set Ts is empty for s = 2 .
Acknowledgment. This work was partly supported by DFG 436 UKR 113/78,
DFG 436 UKR 113/80, INTAS 00-257 and SFB-611 projects. The last two named
authors gratefully acknowledge the hospitality of the Institute of Applied Mathematics
and of the IZKS of the University of Bonn.
1. Besicovitch A. On the sum of digits of real numbers represented in the dyadic systems // Math. Ann. –
1934. – 110. – P. 321 – 330.
2. Eggleston H. G. The fractional dimension of a set defined by decimal properties // Quart. J. Math. Oxford
Ser. – 1949. – 20. – P. 31 – 36.
3. Pratsiovytyi M., Torbin G. Superfractality of the set of numbers having no frequency of n-adic digits,
and fractal probability distributions // Ukr. Math. J. – 1995. – 47, # 7. – P. 971 – 975.
4. Olsen L. Applications of multifractal divergence points to some sets of d-tuples of numbers defined by
their N-adic expansion // Bull. Sci. Math. – 2004. – 128. – P. 265 – 289.
5. Olsen L. Applications of multifractal divergence points to sets of numbers defined by their N -adic ex-
pansion // Math. Proc. Cambridge Phil. Soc. – 2004. – 136, # 1. – P. 139 – 165.
6. Olsen L., Winter S. Normal and non-normal points of self-similar sets and divergence points of self-similar
measures // J. London Math. Soc. – 2003. – 2(67), # 1. – P. 103 – 122.
7. Albeverio S., Pratsiovytyi M., Torbin G. Topological and fractal properties of subsets of real numbers
which are not normal // Bull. Sci. Math. – 2004. – # 208.
8. Albeverio S., Torbin G. Fractal properties of singular continuous probability distributions with independent
Q∗-digits // Ibid. – 2005. – 129, # 4. – P. 356 – 367.
Received 17.06.2005
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