TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

AJMS

1. www.ajms.com


RESEARCH ARTICLE
TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL
CANTOR FUNCTIONS

Yaremenko Mykola Ivanovich
*

*
National Technical University of Ukraine, Igor Sikorsky Kyiv Polytechnic Institute,Kyiv, Ukraine

Corresponding Email: [email protected]

Received: 22-09-2023; Revised: 30-10-2023; Accepted: 16-11-2023

ABSTRACT
In this article, we consider the self-similar generalized Cantor set  ,,,
nn
l
C i C i i
1 , and we
establish the existence of probability true measure  such that   
,..,
1
j
js
EE
s
  



1
01
generated by
n
Ci . The Holder order  of the set 
n
Ci is log
n
s and we establish that      ,,
ll
x n x s i i n

 

   


  
1
1
2
for all not finite n -adic  ,..., .
n
l
x C i i
1
Transcendental numbers, such as e and  are a mathematical expression of nature, we introduce the
transcendental Cantor set generated by transcendental numbers, which can be defined by 
, ,...,
lim
n
kk
k
k
C C C



01
, where the sequence 
k
C is non-increasing and corresponds with the
transcendental number  , for such a set, we consider an analog of the Cantor function.

Keywords: transcendental number, Cantor set, Cantor function, fractal, irregular Cantor set, Holder
continuity.
INTRODUCTION

In recent years there have been several variants of generalization of the Cantor sets [4, 16], the best-
known example of such generalization of the Cantor ternary set is the Smith-Volterra-Cantor sets [4],
which presents the nowhere-dense self-similar set with positive Lebesgue measure. All iterations in the
construction of the Smith-Volterra-Cantor set are self-similar, namely, each step generates the next steps
so that each subinterval is divided in the same ratio. A singular function constructed on the Smith-
Volterra-Cantor set, similar to the Cantor function, is Lipshitz on its domain.

2. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

156
AJMS/Oct-Dec 2023/Volume 7/Issue 4

In this paper, we develop two ways of generalization of self-similar sets, the first is we consider Cantors
sets   ,,,
nn
l
C i C i i
1 with an arbitrary base, for such sets we establish that there exists a uniquely
defined probabilistic measure  defined by   
,..,
1
j
js
EE
s
  



1
01 for all Borel sets E ,and
consider point wise densities for their measure generalized classical results of De-Jun Feng; the second
we construct the irregular or transcendental Cantor set generated by the number e , which we denote by  10Ce
32
such set can be presented as 
, ,...,
10 lim
kk
k
k
C e C C




32
01 , where the collection 
k
C is a
sequence of non-increasing sets corresponded to the number e . For the irregular Cantor sets, we
construct the functions analogous to Smith-Volterra-Cantor functions.

GENERALIZED CANTOR FUNCTION : 0,1 0,1G 

Let 0,1xR then for each natural number n there exists a unique expansion of x in the form of
an infinite series

,...,
.....
k
k
k
ax aa a a
x
n n n n n

     
31 2 4
1 2 3 4
1

Where  
,...
k
k
ax
1 with 0,1,..., 1
k
a x n

Definition 1 The generalized Cantor set   ,,,
nn
l
C i C i i
1 consists of all real numbers 0,1x
, which remain after the removal of all open intervals
1111
, .... , , ... , ......
l l l l
i i i ii i i i
n n n n n n n n
       
    
       
       
1 1 1 1
2 2 2 2


Or the generalized Cantor set 
n
Ci consists of all real numbers 0,1x the n - expansion of
which can be written 
,...,
k
k
k
ax
x
n


1 without a set ,,,
l
ii
1 of numbers 0,1,..., 1
j
in , 1,...,jl
.
Let 0,1x be expressed in basen , we define the Cantor function : 0,1 0,1G  by 

,...,
k
k
k
bx
Gx
s


1
,  0,1,..., 1
k
b x s
for all   ,,,
nn
l
x C i C i i
1 , where 
kk
b x j a defined by   0,1,..., 1, 1,...., 1, 1,..., 1 0,1,..., 1
ll
i i i i n s      
11

for all 
k
ax such that 
n
x C i , and

3. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

157
AJMS/Oct-Dec 2023/Volume 7/Issue 4
   sup ,
n
yx
G x G y y C i



For all0,1 \
n
x C i this definition is correct sinces n l .
Theorem1. For all, 0,1xy , the Holder condition for the generalized Cantor function G given
by G x G y c x y

  

Holds with the best possible constantlog
n
s

THE CUMULATIVE PROBABILITY DISTRIBUTION FUNCTION

For a given general Cantor set 
n
Ci , we define a set  , 0,..., 1
j
js  of functions defined by 1
x
n

0
, 11i
x
nn



1
1
,
……, 11
l
s
i
x
nn




1

Defined for allxR

Example In cases 0C
3 and2C
3 , we have 11
33
x
0
, 21
33
x
1
and 1
3
x
0
, 11
33
x
1 .

Theorem2.For any given general Cantor set   ,...,
n
l
C i i
1 there exists a uniquely defined
probability true measure  such that   
,..,
1
j
js
EE
s
  



1
01

for all Borel sets E . For all continuous functions:g R R , the following equality   
,..,
1
j
jsEE
g x d x g x d x
s
  


01

4. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

158
AJMS/Oct-Dec 2023/Volume 7/Issue 4

For all Borel sets E .
The proof of this theorem is based on the idea that, for all Borel sets ER and any measure  ,
since mapping :    given by   
,..,
1
j
js
EE
s
  


 
1
01

is contractive transformation on  , the mapping :    has a unique fixed point, namely, the
measure  such that  , such measure  satisfies the identity   
,..,
1
j
jsEE
g x d x g x d x
s
  


01

for all continuous functions :g R R .

LOWER AND UPPER DENSITIES OF THE GENERALIZED CANTOR
FUNCTION

The lower and upper  -densities of the measure  at a point x is defined by 

, liminf
2
h
G x h G x h
x
h




  

0

And 

, limsup
2
h
G x h G x h
x
h




  

0
,
Where the function Gx is an extension of the generalized Cantor function given by  
0, 0
, 0 1
1, 1 .
x
G x G x x
x


  



The common value,,xx


    , when such exists, is called the  -density of the measure  at x
.
Definition 2. For0,1x , we define a pair of functions x and x by 

,...,
liminf
kj
k
j
k
ax
x
n




 
1

and   min , 1x x x  
.
Definition 3. We introduce transformation

5. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

159
AJMS/Oct-Dec 2023/Volume 7/Issue 4
     : ,..., ,...,
nn
ll
C i i C i i
11

by 
, 0, ,
1 1 2
, , ,
............
11
, , ,
1 1 2
, , ,
.....,
11
, ,1 .
j j j
j j j
ll
n x n
n x x
n n n
i i i
x n x x
n n n
i i i
n x x
n n n
ii
n x x
nn






   

 
   


    
    
   

     
 
   



   
 
   
1


for all 
,...,
k
k
k
x a x n



1 ,   0,1,..., 1 \ ,,,
kl
a n i i
1 .
Theorem 3. Let 
n
Ci be generalized Cantor set and let 
n
x C i then   
l
x si sn x
 




and  
-
,
;
l
s x is a n finite
x si s x
otherwise
n

 
 








For all not finite n - adic  ,...,
n
l
x C i i
1 we have      ,,
ll
x n x s i i n

 

   


  
1
1
2
.
Proof.
First, we show that assume0,xn



1 and  max , 1x n x t x

   
1 then we obtain    ,
l
x t x t st si snx



   
.
Indeed, if  max ,
l
i
x n x t x
n

   
1 then  0, ,n x t x t

   

1
so

6. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

160
AJMS/Oct-Dec 2023/Volume 7/Issue 4
  
 
 
, 1
l
x t x t
si snx
st s st




  
;
if 1
l
i
x r x
n
    then   

 
11
,
11
l
ss
sr
x t x t
ss
si snx
st ss
s x r s x
ss


  







   
   
  
   
   
,

where we used the following estimation 0,
t
tt
s




holds for each0,1t .
Therefore, for each 
n
x C i and all  0,tn


1 , we can choose the sequence 
d
j such that lim ... 0,1
k
jj
k
x


1
, we take  lim ...
k
jj
k
yx




11
1 so that  ...
k
jj
yx



11
1 and 
k
yx


1
where mapping k

1 is the 1k -th iteration of mapping  . Therefore, we have    ... , ,
k
jj
x t x t y t y t


    

11
1

for all k
t tn


1 . So, we have   

 ,,
k
x t x t n y t y t



     

1 hence   

 ,,
k
x t x t n y t y t



     

1
.
Since 
k
yx


1 and 11
k
yx

   
1 we have   

    
 
,
liminf
min liminf , liminf 1
,
t
kk
l
kk
l
x t x t
st
si sn x x
si sn x







 




     

0

that proves the theorem.

IRREGULAR OR TRANSSENDERNTAL CANTOR SETS

7. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

161
AJMS/Oct-Dec 2023/Volume 7/Issue 4

In this chapter, we introduce new class sets, which are uncountable, compact, perfect, and totally
disconnected. Such sets are closely related to the Cantor and Smith-Volterra sets, however, their nature is
completely different from the regular fractal sets, so we will call this class irregular fractals.
As direct corollaries of the Hermite-Weierstrass theorem, we obtain that the numbers e and  are
transcendental, and we write expansions of e and  -2 -3 -4 -5 -6 -7 -8
10 2×10 +7×10 +1×10 +8×10 + 2×10 +8×10 +1×10 +...e


2

and -2 -2 -3 -4 -5 -6 -7 -8
10 = 3×10 +1×10 + 4×10 +1×10 +5×10 +9×10 + 2×10 +.. .
.
By using the transcendental number e , we construct the irregular Cantor sets from the unit compact
interval 0,1C
0 by performing the recursive process: the first iteration consists of the removal of the
open interval 22
1 2 1 2
,
2 2×10 2 2×10



 from the interval 0,1 so that the remaining set is 22
1 2 1 2
0, ,1
2 2×10 2 2×10
C
   
   
   
   
1
;
the second iteration is removing the subintervals of the common width -3
7×10 from the middle of each
of the two remaining intervals, so for the second step we leave the set 2
22
22
2
1 1 2 7
0,
2 2 2×10 4×10
1 1 2 7 1 2
,
2 2 2×10 4×10 2 2×10
1 2 1 1 2 7
,1
2 2×10 2 2 2×10 4×10
1 1 2 7
1 ,1 ;
2 2 2×10 4×10
C

   



    


 
     



   


2 3
3
3
3

employing expansions of the number e , we continue the process indefinitely and obtain a sequence 
k
C
of non-increasing sets k
C such that kk
CC


1 , the set of points, that remain after infinite numbers
of iterations, is called the irregular Cantor set generated by the number e and denoted by  10Ce
32 .
We can write
, ,...,
10 lim
kk
k
k
C e C C




32
01 .

Applying a similar process for the expansion of the number  , we obtain the irregular Cantor set
generated by the number  and denoted by  10C 
32 .
From the definitions, we have that

8. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

162
AJMS/Oct-Dec 2023/Volume 7/Issue 4
  10 1 10
L
C e e


3 2 2

and
  10 1 10
L
C  


3 2 2
.
The coefficient is a 10
2 is not important we can use any suitable coefficient, which guarantees the
correctness of the iteration process.
The irregular Cantor sets  10Ce
32 and  10C 
32 are similar to the ternary Cantor set in the sense
that we are removing the part of the middle sets at each step of the iteration process, however, the nature
of irregular sets is different from classical Cantor and Smith-Volterra-Cantor since they are regular in the
sense that k -iteration depends systematic on 1k -iteration, on the contrary, the width of removed
intervals in the irregular set is prescribed by the nature of the generated number is unregular.
The interior of the irregular Cantor set is empty, namely, it does not contain any interval, which is open in
the standard topology of the real line. The irregular Cantor sets are not self-similar since, at each step of
the iteration, the set of numbers fed by expansion -2 -3 -4 -5 -6 -7 -8
10 2×10 +7×10 +1×10 +8×10 + 2×10 +8×10 +1×10 +...e


2
is unique.

PROPERTIES OF IRREGULAR CANTOR SETS AND IRREGULAR CANTOR
FUNCTIONS

The irregular Cantor sets are uncountable, closed,and totally disconnected, with uniquely defined
Lebesgue measure  10
n
L
C  


2 for the irregular Cantor set 
n
C generated by the number  ,
where the number n determines the base as was explained earlier. The proofs of these statements are
similar to proofs for the classical Cantor set.
Based on the irregular Cantor set 10Ce
32 , we define the irregular Cantor function : 0,1 0,1F 
by iteration procedure as follows: 
22
0 0,
1 1 2 1 2
,,
2 2 2×10 2 2×10
1 1 ,
if x
F x if x
if x


 
    

 
1

9. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

163
AJMS/Oct-Dec 2023/Volume 7/Issue 4

22
22
22
0 0,
1 1 1 2 7 1 1 2 7
,,
4 2 2 2×10 4×10 2 2 2×10 4×10
1 1 2 1 2
,,
2 2 2×10 2 2×10
3 1 1 2 7 1 1 2 7
1 ,1 ,
4 2 2 2×10 4×10 2 2 2×10 4×10
1 1 ,
if x
if x
F x if x
if x
if x


   

    
   
    

 
    


   
          
   


33
2
33

et cetera for all  0,1 \ 10x C e


32 , and, by continuity, we put function irregular Cantor function F
linear on  10Ce
32 .
The irregular Cantor function : 0,1 0,1F  is a continuous monotone function, therefore, there
exists the derivative F of F for Lebesgue almost all 0,1x . We have 0Fx for all  0,1 \ 10Ce
32
and F x const for all    
, ,....
0,1 \ 10 0,1
k
k
Ce 


 
 


32
12 , where points k

are points of division such as 2 2 2
1 1 2 7 1 1 2 7 1 2
, , ,....
2 2 2×10 4×10 2 2 2×10 4×10 2 2×10
   
    
   
   
33
. At each division point,
there are two derivative numbers: one equals zero, and the second is a positive constant.

REFERENCES

1. E. Ayer and R. S. Strichartz, Exact Hausdorff measure and intervals of maximum density for
Cantor sets, Trans. Amer. Math. Soc. 351, N. 9, (1999), 3725-3741.
2. I. Banic, G. Erceg, S. Greenwood, J. Kennedy, Transitive points in CR-dynamical systems,
Topology and its Applications, V. 326, (2023).
3. T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem of fractal sets of real and
integers, Proc. London Math. Soc. 64, N. 3, (1992), 95 -124.
4. D. A. Bresoud, Radical Approach to Lebesgue’s Theory. Cambridge University Press. New York
(2008).
5. D. R. Dellwo, Fat Cantor sets and their skinny companions, HeliyonVol. 9, Issue 5, Published
online: March 31, (2023).
6. F.M. Dekking, W. Li, How smooth is a devil’s staircase?, Fractals 11 (1), (2003), 101–107.

10. TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

164
AJMS/Oct-Dec 2023/Volume 7/Issue 4

7. Cornean, H., Herbst, I.W., Møller, J. et al. Singular Distribution Functions for Random Variables
with Stationary Digits. MethodolComput Appl Probab 25, 31, (2023).
8. David R. Dellwo, Fat Cantor sets and their skinny companions, Heliyon, V. 9, Issue 5, (2023).
9. K. Falconer, Wavelet transforms and order-two densities of fractals, J. Statist. Phys. 67. (1992),
781-793.
10. A. K. Golmankhaneh and D. Baleanu, New heat and Maxwell’s equations on Cantor cubes Rom.
Rep. Phys. (2017), 69, 109.
11. T.Y. Hu, K.S. Lau, Fourier asymptotics of Cantor type measures at infinity, Proc. Amer. Math.
Soc. 130, (9), (2002), 2711–2717.
12. W. Li, D. Xiao, F.M. Dekking, Non-differentiability of devil’s staircase and dimension of subsets
of Moran sets, Math. Proc. Cambridge Philos. Soc. 133, (2), (2002), 345–355.
13. P. Nowakowski, The Family of Central Cantor Sets with Packing Dimension Zero, Tatra
Mountains Mathematical Publications, vol.78, N.1, (2021), 1-8.
14. F. Prus-Wisniowski, F. Tulone, The arithmetic decomposition of central Cantor sets. J. Math.
Anal. Appl. 467, (1), (2018), 26–31.
15. R. S. Strichartz, A. Taylor, and T. Zhang, Densities of self-similar measure on the line,
Experiment. Math. 4, N. 2, (1995), 101-128.
16. Z. Y. Wen & J.Wu, Haudorff Dimension of Homogeneous Perfect Sets Acta Math. Hungar. 107
(1-2) (2005), 35–44.