In 1883, G. Cantor constructed the ﬁrst widely known

example of a perfect nowhere-dense fractal set in the real

line, similarly, sets were constructed by V. Volterra and

J. Smith in 1881 and 1875, respectively. The perfect

nowhere-dense fractal sets are the base for the construc-

tion of many counterexamples in analysis such as sin-

gular functions, which are Holder continuous with some

degree but not with the others. The Cantor sets have a

self-similar, fractal structure since the Cantor set is equal

to two of its translated copies after being shrunk by fac-

tor three. Since this branch of analysis is intensively

developing, there is extensive literature on the subject

that we will not review in this short article, some works

are given in references [1 - 8].

The main goal of this article is to introduce and study

a new type of perfect nowhere-dense sets, which are not

self-similar, and correspondent singular non-self-similar

functions.

For any x∈[0,1], there exists a unique decimal ex-

pression xin the form of an inﬁnite series

x=X

k=1,...,

10k=a1

101+a2

102+a3

103+a4

104+..... (1)

where a set {ak}of numbers ak∈ {0,1, ..., 9}, therefore,

eachx∈[0,1] can be uniquely presented by inﬁnite dec-

imal series x=Pk=1,...,

ak(x)

10kwith a unique set {ak(x)}

of numbers ak(x)∈ {0,1, ..., 9}.

The numbers eand πare transcendental, and the

numbers eand πcan be written in the form of expansions

e= 2+7×10−1+1×10−2+8×10−3+2×10−4+... (2)

and

π= 3×10+1×10−1+4×10−2+1×10−3+5 ×10−4+..,

(3)

therefore, the numbers eand π, we identify with se-

quences

e↔ {2,7,1,8,2,8,1, ...}={˜ake}k=1,.... (4)

and

π↔ {3,1,4,1,5,9,2, ...}={˜akπ}k=1,.... ,(5)

respectively.

Deﬁnition 1. The irregular Cantor sets

Ce([0,1]) and Cπ([0,1]) consist of all real num-

bers x∈[0,1] such that

xe=X

k=1,...,

ake

10k=a1e

101+a2e

102+a3e

103+a4e

104+... (6)

and

xπ=X

k=1,...,

akπ

10k=a1π

101+a2π

102+a3π

103+a4π

104+... (7)

where ake(x)∈ {0,1, ..., 9} \˜akeand akπ(x)∈

{0,1, ..., 9} \˜akπfor each k, respectively.

Irregular Cantor sets have the following properties:

1. The Cantor sets Ce([0,1]) and Cπ([0,1]) have a

cardinality of the continuum.

2. The Cantor sets Ce([0,1]) and Cπ([0,1]) are

closed in the topology of the real line.

The irregular Cantor sets Ce ([0, 1]) and Cπ ([0, 1]), and the Cantor-

Lebesgue irregular functions Ge and Gπ

MYKOLA YAREMENKO

The National Technical University of Ukraine, Kyiv 04213, Kyiv, UKRAINE

Abstract: - In this article, we introduce and study a new class of perfect nowhere-dense sets, which are not self-

similar in any subset, also, we constructed the correspondent singular functions. We construct a two-

dimensional irregular Cantor set Ce,π ([0, 1]) on the real plane.

Keywords: - Cantor set, Cantor function, irregular Cantor set, irregular Cantor function, fractal.

Received: September 26, 2022. Revised: July 21, 2023. Accepted: August 17, 2023. Published: September 15, 2023.

1. Introduction

2. Generalized irregular Cantor sets

PROOF

DOI: 10.37394/232020.2023.3.5

Mykola Yaremenko

E-ISSN: 2732-9941

Volume 3, 2023

3. The Cantor sets Ce([0,1]) and Cπ([0,1]) are

compact in the topology of the real line.

4. The Cantor sets Ce([0,1]) and Cπ([0,1]) are rare

in the topology of the real line.

Theorem 1. The Lebesgue measures of the

Cantor sets Ce([0,1]) and Cπ([0,1]) equal zero.

Proof.

The Lebesgue measure µof the Cantor sets Ce([0,1])

and Cπ([0,1]) is given by

µ(Ce([0,1])) = µ([0,1]) −µ([0,1] \Ce([0,1])) =

= 1 −9

10 +9

10 1

10 2+...=

= 1 −9

10 Pk1

10 k= 0.

Theorem 1 is proven.

The irregular Cantor set Ce([0,1]) can be con-

structed from the unit compact interval C0e= [0,1] by

performing the following recursive procedure, which is

based on correspondence e↔ {2,7,1,8,2,8,1, ...}=

{˜ake}k=1,....: the ﬁrst iteration consists of the removal of

the open interval 2

10 ,3

10 corresponding to a1e= 2,the

second iteration consists of the removal of nine subinter-

vals corresponding to a2e= 7,the second iteration con-

sists of the removal eighty-one subintervals correspond-

ing to a3e= 1,and so on. Continuing this inﬁnite proce-

dure, we obtain a sequence {Cke}of strictly decreasing

sets Cksuch that Ck⊃Ck+1 with strict inclusions for

all numerators, the limited set of points, that remain

after inﬁnite numbers of iterations, coincides with an ir-

regular Cantor set Ce([0,1]) = lim

k→∞

Ck=Tk=0,1,..., Ck

. Similar considerations can be accomplished for a set

Cπ([0,1]) taking π↔ {3,1,4,1,5,9,2, ...}as a

guiding sequence.

We are going to introduce the concept of the Cantor-

Lebesgue irregular functions Geand Gπfor the irregular

Cantor sets Ce([0,1]) and Cπ([0,1]) through an itera-

tive procedure. We will construct the function Ge, the

function Gπcan be constructed similarly.

We present the ordinate axis in base nine as

y=X

k=1,...,

9k=b1

91+b2

92+b3

93+b4

94+..... (8)

where a set {bk}of numbers bk∈ {0,1, ..., 8}.

We construct the sequence {φn}of continuous

monotone-increasing functions φn, n = 1,2, , ....,

which converges to the function Ge. The continu-

ous monotone-increasing function φ1takes the constant

value 2

9on the open interval 2

10 ,3

10 , which corresponds

with the 2 in the expansion for eand has been removed

from [0,1], and the continuous monotone-increasing

function φ1is linear on the remaining intervals. The

function φ2is given by

φ2=











92,7

102,8

102,

9+7

92,1

10 +7

102,1

10 +8

102,

9,2

10 ,3

10 ,

9+7

92,3

10 +7

102,3

10 +8

102,

9+7

92,4

10 +7

102,4

10 +8

102,

9+7

92,5

10 +7

102,5

10 +8

102,

9+7

92,6

10 +7

102,6

10 +8

102,

9+7

92,7

10 +7

102,7

10 +8

102,

9+7

92,8

10 +7

102,8

10 +8

102,

linear on remaining intervals;

which corresponds with the 7, in the expansion for e. We

continue this procedure and obtain the sequence {φn}of

continuous monotone-increasing functions.

For functions φn, we have the uniform estimation

|φn+1 (x)−φn(x)| ≤ 9n

for all x∈[0,1]. So that the sequence {φn}converges

to the continuous monotone-increasing function which

we denote Ge.

Deﬁnition 2. The limit of the sequence {φn}

deﬁned as about is called the Cantor-Lebesgue ir-

regular function Ge.

Similarly, employing expansion of πwe can de-

ﬁne the Cantor-Lebesgue irregular continuous monotone-

increasing function Gπ.

Straightforward considerations yield the following

statements.

Lemma 1. The Cantor-Lebesgue irregular

functions Geand Gπhave the following proper-

ties:

1) Geand Gπare continuous and monotone-

increasing functions however they are not abso-

lutely continuous;

2) Geand Gπare singular functions;

3) Geand Gπmap the Cantor sets Ce([0,1])

and Cπ([0,1]) onto [0,1].

Theorem 2. The Cantor-Lebesgue irregular

functions Ge(x)and Gπ(x)are locally concave-

convex at point ˜xif and only if

˜x∈[0,1] \Ce([0,1]) and ˜x∈[0,1] \Cπ([0,1]), re-

spectively.

Proof. We are going to prove the theorem for the

Ge; in case of the Gπ, a similar consideration can be

employed.

We denote ΘGethe set of all points where the function

Getakes constant values. If x∈ΘGethen Ge(x) is

concave-convex in a neighborhood Υ of x. Let [0,1] \Θ

and let Gebe concave-convex in a neighborhood of the

point x. There are points y, z ∈[0,1] \Θ and a positive

constant δsuch that y < z,

(y−δ, z −δ)⊂Υ,

and (y, z)⊂Θ. We have that exist points ˜y, ˜zsuch

that

˜y∈(y−δ, y)∩([0,1] \Θ)

3. The Cantor-Lebesgue irregular

Functions

PROOF

DOI: 10.37394/232020.2023.3.5

Mykola Yaremenko

E-ISSN: 2732-9941

Volume 3, 2023

and

˜z∈(z−δ, z)∩([0,1] \Θ) ,

and we apply a standard argument since Geis the

increasing function, the pair of straight lines pass

through points (˜y, Ge(˜y)), y+ 2−1z, Gey+ 2−1z

and (˜z, Ge(˜z)), z+ 2−1y, Gez+ 2−1y, respec-

tively, then the straight line passes through points

(˜y, Ge(˜y)), y+ 2−1z, Gey+ 2−1z is passing under

point (y, Ge(y)), and the straight line passes through

points (˜z, Ge(˜z)), z+ 2−1y, Gez+ 2−1y is passing

over point (z, Ge(z)) , therefore, the restriction of Ge

to (˜y, ˜z) is not concave-convex, this contradiction proves

the statement of the theorem.

Lemma 2. The Cantor-Lebesgue irregular

functions Geand Gπsatisfy the Banach condi-

tions T1.

The proof is straightforward.

Now, we are going to construct the irregular analog

of the Sierpinski carpet, a fractal-like two-dimensional

structure produced by an irregular iterated system with

a constant base of ten.

We start with a unit square [0,1] ×[0,1] and present

its points (x, y)∈[0,1] ×[0,1] in form of an expansion

in base ten

x=X

k=1,...,

10k=a1

101+a2

102+a3

103+a4

104+.....

y=X

k=1,...,

10k=b1

101+b2

102+b3

103+b4

104+......

We assume that abscissa and ordinate axes govern by

numbers eand π, respectively. A two-dimensional irreg-

ular Cantor set Ce,π ([0,1]) is constructed by an iter-

ative procedure of deleting squares, whose coordinates

expressed in the base ten do not both have the respec-

tive index kdigits of respective numbers eand π, respec-

tively (x, y)∈[0,1] ×[0,1]. The ﬁrst deleted square is

2

10 ,3

10 ×3

10 ,4

10 , next we delete 99 squares governing

by 1 and 1 from eand π, respectively. The process is

inﬁnite removing squares, and as a result, we obtain a

fractal irregular two-dimensional set, which we called a

two-dimensional irregular Cantor set Ce,π ([0,1]).

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4. Two-dimensional irregular

Cantor set Ce,π ([0, 1])

References

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The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

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Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

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(Attribution 4.0 International, CC BY 4.0)

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PROOF

DOI: 10.37394/232020.2023.3.5

Mykola Yaremenko

E-ISSN: 2732-9941

Volume 3, 2023