On one fractal property of the Minkowski function

0.1 MB·

by  Symon Serbenyuk

ON ONE FRACTAL PROPERTY OF THE MINKOWSKI
FUNCTION
SYMON SERBENYUK
Abstract. Thearticleisdevotedtoanswerthequestionaboutpre-
7 servingtheHausdorff-BesicovitchdimensionbythesingularMinkowski
1 function. ItisprovedthatthefunctionisnottheDP-transformation,
0
i.e.,theMinkowskifunctiondoesnotpreservetheHausdorff-Besicovitch
2
dimension.

n
a
J
It is well-known that the main problem of the fractals theory is a
3
problem of calculating of dimension. In particular, the dimension is the
2
Hausdorff-Besicovitch dimension (a fractal dimension). But, since some

]
A classes of sets have complicated determination, a calculation of a value
C of the dimension is a difficult and labour-consuming problem for these
. sets. Therefore, to simplify of fractal dimension calculation, a problem
h
of searching of auxiliary facilities appears. Transformations preserving
t
a
the Hausdorff-Besicovitch dimension (DP-transformations) are such fa-
m
cilities. The transformations help to simplify of sets determinations and
[
to investigate of belonging to the class of DP-tranformations of other
1
transformations.
v
0 Monotone singular distribution functions as transformations of the
4
segment [0;1] are very interesting for studying of DP-transformations.
6
6 The present article is devoted to considering of fractal properties of one
0 example of such functions. In the article a preserving of the Hausdorff-
.
1 Besicovitch dimension bytheMinkowski functionisinvestigated. Results
0
of the present article were presented by the author of the article on Sec-
7
1 ond Interuniversity Scientific Conference on Mathematics and Physics
v: for Young Scientists in April, 2011 [6].
i The following function
X
r G(x) = 21−a1 −21−(a1+a2) +...+(−1)n−121−(a1+a2+...+an) +...,
a
is called the Minkowski function. An argument x of the function deter-
mined in terms of representation of real numbers by continued fractions,
2010 Mathematics Subject Classification. 28A80, 26A30, 11K55,03E99,11K50.
Key words and phrases. Function with complicated local structure, singular func-
tion, fractal, self-similar set, Hausdorff-Besicovitchdimension, Minkowski function.
1
2 SYMON SERBENYUK
i. e.,
1
[0;1] ∋ x = ≡ [0;a ,a ,...,a ,...], a ∈ N.
1 2 n n
1
a +
1
1
a +
2 a +...
3
To establish of the one-to-one correspondence between rational numbers
and quadratic irrationalities, the function was introduced by Minkowski
in[5]. AdifficultyofinvestigationofpreservingtheHausdorff-Besicovitch
dimension by the Minkowski function is caused by singularity and a com-
plexity of the argument determination of the function.
Let us consider the following set
E ≡ {x : x = [0;a ,a ,...,a ,...],a ∈ {1,2,3,...,9} ∀i ∈ N}.
9 1 2 i i
”The geometry of continued fractions” does not have a property of
”classical self-similarity”. It is the reason of rather greater difficulties for
obtaining exact results of fractal properties of continued fractions sets.
The main researches, in which fractal properies of some types of sets
of continued fractions are studied, there are the articles of Jarnik [4]
(the sets E , that representation of its elements by continued fractions
n
contains symbols that do not exceed n), Good [1] (the sets of contin-
ued fractions, whose elements a (x) quickly tend to infinity), Hirst [3]
n
(the sets of such continued fractions, whose elements belong to infinite
sequence of positive integers and increase indefinitely) and in [2] Hensley
clarified estimations for Hausdorff-Besicovitch dimension of E such that
n
were obtained by Jarnik and Good:
1 1
1− ≤ α (E ) ≤ 1− ,n > 8.
nlg2 0 n 8nlgn
Whence, for E we obtain that
9
0,6308969 ≤ α (E ) ≤ 0,985445112. (1)
0 9
The answer to the main question of this article follows from the next
statement about a value of the Hausdorff-Besicovitch dimension of the
set G(E ).
9
Theorem 1. A value of the Hausdorff-Besicovitch dimension of the fol-
lowing set
2 2 2(−1)n−1
y : y = G(x) = − +...+ +...,x = [0;a ,a ,...,a ,...] ,
2a1 2a1+a2 2a1+a2+...+an 1 2 n
(cid:26) (cid:27)
FRACTAL PROPERTIES OF THE MINKOWSKI FUNCTION 3
where ∀i ∈ N : a ∈ {k ,k ,...,k } and {k ,k ,...,k } is a fixed tuple
i 1 2 S 1 2 S
of positive integers for a fixed positive integer number S > 1, can be
calculate by the formula:
1 α0 1 α0 1 α0 1 α0
+ + +...+ = 1.
2k1 2k2 2k3 2kS
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Proof. Let us write by ∆ =
0
2 2 2(−1)n−1 2 (−1)n (−1)n+1
= y : y = − +...+ +...+ + +...
2a1 2a1+a2 2a1+a2+...+an 2a1+a2+...+an 2an+1 2an+1+an+2
(cid:26) (cid:18) (cid:19)(cid:27)
the set
2 2 2(−1)n−1
∆ ≡ y : y = G(x) = − +...+ +... .
0 2a1 2a1+a2 2a1+a2+...+an
(cid:26) (cid:27)
Call the set
(−1)n (−1)n+1 (−1)n+2
D ≡ y : y = + + +...
n 2an+1 2an+1+an+2 2an+1+an+2+an+3
(cid:26) (cid:27)
by an ”indicative set” of rank n.
Let S = 2 and k < k , then 2 > 2 ,
1 2 2k1 2k2
2 2 2 2
sup∆ = − + − +... =
0 2k1 2k1+k2 2k1+k2+k1 2k1+k2+k1+k2
2 2 2(2k2 −1)
= 2k1 − 2k1+k2 = ,
1− 1 1− 1 2k1+k2 −1
2k1+k2 2k1+k2
2 2 2 2
inf∆ = { − + − +... =
0 2k2 2k2+k1 2k2+k1+k2 2k2+k1+k2+k1
2 2 2(2k1 −1)
= 2k2 − 2k1+k2 = .
1− 1 1− 1 2k1+k2 −1
2k1+k2 2k1+k2
So,
2 2k2 −2k1
λ(∆ ) = ,
0 2k1+k2 −1
(cid:0) (cid:1)
where λ(·) is a diameter of a set.
Consider the following two sets
2 2 2(−1)n−1
∆ = y : y = − +...+ +...
kj1 2kj1 2kj1+a2 2kj1+a2+...+an
(cid:26) (cid:27)
where j ∈ {1,2} and {k ,k } ∋ k is a fixed for ∆ . That is
1 1 2 j1 kj1
2 1 1 1 2 2
∆ = y : y = 1− + − +... = − D ,
k1 2k1 2a2 2a2+a3 2a2+a3+a4 2k1 2k1 1
(cid:26) (cid:18) (cid:19)(cid:27) (cid:26) (cid:27)
4 SYMON SERBENYUK
2 1 1 1 2 2
∆ = y : y = 1− + − +... = − D .
k2 2k2 2a2 2a2+a3 2a2+a3+a4 2k2 2k2 1
(cid:26) (cid:18) (cid:19)(cid:27) (cid:26) (cid:27)
It is easy to see that
2 1
λ(∆ ) = λ(D ) = λ(∆ )
k1 2k1 1 2k1 0
and
2 1
λ(∆ ) = λ(D ) = λ(∆ ).
k2 2k2 1 2k2 0
Inthesecondstepweobtainthefollowingfoursets: ∆ ,∆ ,∆ ,∆ .
k1k1 k1k2 k2k1 k2k2
In the nth step we shall have 2n sets
n 2(−1)m−1 ∞ 2(−1)t−1
∆ ≡ y : y = + ,
kj1kj2...kjn ( 2kj1+kj2+...+kjm 2kj1+kj2+...+kjn+an+1+...+at)
m=1 t=n+1
X X
where k ,k ,...,k is a fixed tuple of numbers from {k ,k }, and the
j1 j2 jn 1 2
following expression is true for all n ∈ N.
λ ∆
kj1kj2...kjn−1kjn = 1 ∈ 1 , 1 ,
λ(cid:16) ∆ (cid:17) 2kjn 2k1 2k2
kj1kj2...kjn−1 (cid:26) (cid:27)
(cid:16) (cid:17)
The last-mentioned fact follows from the next proposition.
Proposition 1. The condition
λ(D )
n+1
= 1
λ(D )
n
holds for all n ∈ N.
Really, let n = 2l, l ∈ N. Then
1 1 1
D = D ≡ y : y = − + − +... ,
n+1 2l+1 2a2l+2 2a2l+2+a2l+3 2a2l+2+a2l+3+a2l+4
(cid:26) (cid:27)
1 1 1 1
D = D ≡ y : y = − + − +...
n 2l 2a2l+1 2a2l+1+a2l+2 2a2l+1+a2l+2+a2l+3 2a2l+1+a2l+2+a2l+3+a2l+4
(cid:26) (cid:27)
1 1 1−2k1
sup(D ) = − 2k2 + 2k1+k2 = ,
2l+1 1− 1 1− 1 2k1+k2 −1
2k1+k2 2k1+k2
1 1 1−2k2
inf(D ) = − 2k1 + 2k1+k2 = ,
2l+1 1− 1 1− 1 2k1+k2 −1
2k1+k2 2k1+k2
2k2 −2k1
λ(D ) = .
2l+1 2k1+k2 −1
FRACTAL PROPERTIES OF THE MINKOWSKI FUNCTION 5
Similarly,
1 1 2k2 −1
sup(D ) = 2k1 − 2k1+k2 = ,
2l 1− 1 1− 1 2k1+k2 −1
2k1+k2 2k1+k2
1 1 2k1 −1
inf(D ) = 2k2 − 2k1+k2 = ,
2l 1− 1 1− 1 2k1+k2 −1
2k1+k2 2k1+k2
2k2 −2k1
λ(D ) = .
2l 2k1+k2 −1
So, λ(D ) = λ(D ). A proof of the equality is analogical for the
2l+1 2l
case of n = 2l+1.
Let I be a segment, whose endpoints coincide with end-
kj1kj2...kjn−1kjn
points of the corresponding set ∆ . Since ∆ ⊂ I and ∆ is
kj1kj2...kjn−1kjn 0 0 0
a perfect set and
∆ = [I ∩∆ ]∪[I ∩∆ ],
0 k1 0 k2 0
[I ∩∆ ] 2−∼k1 ∆ , [I ∩∆ ] 2−∼k2 ∆ ,
k1 0 0 k2 0 0
the theorem proved for the case of S = 2.
Let S > 2 and k < k < ... < k . Similarly, in the nth step we shall
1 2 S
have the Sn sets ∆ . It is easy to see that
kj1kj2...kjn
λ(∆kj1kj2...kjn−1kjn) = 1 ,
λ(∆kj1kj2...kjn−1) 2kjn
where k ∈ {k ,k ,...,k },S ∈ N,S > 1.
jn 1 2 S
Since the set ∆ is a compact self-similar set of the space R1, the
0
Hausdorff-Besicovitch dimension α (∆ ) of the set ∆ is calculating by
0 0 0
the formula:
1 α0 1 α0 1 α0 1 α0
+ + +...+ = 1.
2k1 2k2 2k3 2kS
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:3)
The theorem is proved.
So, for k = 1,k = 2,...,k = 9 we obtain that
1 2 9
α (G(E )) ≈ 0,9985778625536. (2)
0 9
From(1)and(2)itfollowsthatα (E ) 6= α (G(E )). So,theMinkowski
0 9 0 9
function does not preserve the Hausdorff-Besicovitch dimension.
6 SYMON SERBENYUK
References
[1] G.T.Good,Thefractionaldimensionaltheoryofcontinuedfractions,Proc. Cam-
bridge Phil. Soc. 37, 199–228(1941).
[2] D. Hensley, Continued fraction Cantor sets, Hausdorff dimension and functional
analysis, Journal of Number Theory 40, 336–358 (1992).
[3] K. E. Hirst, Fractional dimension theory of continued fractions, Quart. J. Math.
21, 29–35 (1970).
[4] V. Jarnik, Zur metrichen Theorie den diophantischen Approximationen, Recueil
Math. Moscow 36, 371–382(1929).
[5] H. Minkowski, Gesammeine Abhandlungen, Berlin 2, 50–51 (1911).
[6] S.O.Serbenyuk,PreservingofHausdorff-Besicovitchdimensionbythe monotone
singular distribution functions, Second Interuniversity Scientific Conference on
Mathematics and Physics for YoungScientists: Abstracts,Kyiv,2011,P.106-107.
(in Ukrainian) Link: https://www.researchgate.net/publication/301637057
Institute of Mathematics, National Academy of Sciences of Ukraine,
3 Tereschenkivska St., Kyiv, 01004, Ukraine
E-mail address: [email protected]; [email protected]