Dimensions of modified Besicovitch subsets of Moran fractal

Chaos, Solitons and Fractals 42 (2009) 2779–2785

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Dimensions of modified Besicovitch subsets of Moran fractal q Wu Yahao *, Wu Min School of Mathematical Sciences, South China University of Technology, Guangzhou 510641, People’s Republic of China

a r t i c l e

i n f o

a b s t r a c t

Article history: Accepted 31 March 2009

We determine the Hausdorff dimension of modified Besicovitch subsets of some Moran fractal. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction P xn For any x 2 ½0; 1Þ, let x ¼ 1 n¼1 2n denote the unique nonterminating 2-ary expansion of x, where xn 2 f0; 1g. Borel proved in 1909 that for Lebesgue almost all x 2 ½0; 1Þ, the frequencies of all digits appears equally. This is the famous Borel’s Theorem on normal numbers [7]. A natural problem is to investigate the set of numbers x in unit interval for which the frequencies of all digits do not appear equally. That is the set:

( BðpÞ ¼

x 2 ½0; 1Þ : lim

n!1

n 1X 1j ðxk Þ ¼ pj ; j; xk 2 f0; 1; . . . ; m  1g n k¼1

)

   where p ¼ ðp0 ; p1 ; . . . ; pm1 Þ – m1 ; m1 ; . . . ; m1 is a probability vector, 1j ðÞ is the characteristic function of singleton fjg and m is a positive integer larger than 2. Besicovitch [2] consider the size of the set from the pointview of dimension in 1934. He studied the Hausdorff dimension of BðpÞ for m ¼ 2 and proved that

dimH BðpÞ ¼

p log p  ð1  pÞ logð1  pÞ log 2

As for the general case, Eggleston [9] got the following result in 1949 for m-ary expansions:

P  m1 j¼0 pj log pj dimH BðpÞ ¼ log m Henceforth, the set BðpÞ is called Besicovitch set. In recent years, Barreira [1], Bisbas [5] and Olsen[19] et al. have made some research related to the the frequencies of numbers. The results they got show that the Hausdorff dimensions of the sets considered in their paper have relation to the properties of the m-ary numbers. Furthermore, Xie et al. extended the results about Besicovitch sets to a more general case which are called the modified Besicovitch–Eggleston sets. The detailed definition and results can be found in [23].

q Research supported by the National Science Foundation of China (10571063, 10631040), the National Science Foundation of Guangdong Province of China (05006515). * Corresponding author. E-mail addresses: [email protected] (W. Yahao), [email protected] (W. Min).

0960-0779/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.188

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On the other hand, Ma et al. [12] considered the Besicovitch subsets of self-similar sets which defined as follows: Given an integer m P 2, let f/0 ; /1 ; . . . ; /m1 g be similarity contracting with similarity ratio fr0 ; r1 ; . . . ; rm1 g, and let E be the self-similar set of the family of the similarities with the open set condition(OSC) [10] satisfied. Furthermore, assume that N f/j ð½0; 1Þgm1 j¼0 be a collection of disjoint sub-intervals of [0,1] and p be the coding mapping from f0; 1; . . . ; m  1g to E, then ! for a given probability vector ðp0 ; p1 ; . . . ; pm1 Þ, the Besicovitch subset Eð p Þ of self-similar set E is defined as:

! Eð p Þ ¼

(

n 1X pðxÞ 2 E : n!1 lim 1j ðxk Þ ¼ pj ; x 2 f0; 1; . . . ; m  1gN ; 0 6 j 6 m  1 n k¼1

)

Pm1 pj log pj ! ! One of the results we can get from [4] is that dimH Eð p Þ ¼ Pj¼0 and the Hausdorff measure of Eð p Þ under this dimenm1 p log r j j¼0 j sion is infinity. In this paper, we extend the Besicovitch subsets of self-similar sets to a more general case, which we call it the modified besicovitch subset of Moran fractal(defined in next section) and get some dimension results, which show that the Hausdorff dimensions of these sets have relation not only to the properties of the m-ary numbers, but also to the partition of N. 2. Notations and main results S Let J ¼ ½0; 1 be the unit interval. D ¼ f0; 1; . . . ; m  1gðm P 2Þ, D ¼ kP0 Dk with D0 ¼ /, Dk ¼ fx ¼ x1 x2 . . . xn : xi 2 D; 1 6 i 6 ng, DN ¼ fx ¼ x1 x2 . . . : xi 2 Dg. Let fJ x : x 2 Dk gðJ / ¼ JÞ be the non-empty compact subsets of J such that: T S For any k P 0; x 2 Dk ; J xi \ J xj ¼ /; i–j and jJjJxix jj ¼ ri , with 0 < r i < 1; i 2 D, then the set F ¼ kP0 Jx 2Dk J x is called a Moran fracN tal [8]. Suppose that w is the projecting mapping from D to F, that is, for any point a 2 F, there exists x ¼ x1 x2    xn    2 DN such that a ¼ wðxÞ ¼ \nP1 J x1 x2 xn . S Let M k ðk ¼ 1; 2; . . . ; lÞ be a partition of N, i.e. N ¼ lk¼1 M k , where M k ¼ fmk;1 ; mk;2 ; . . .g, mk;1 < mk;2 < . . . and T kg exists and is positive throughout Mi M j ¼ /; i–j. Suppose the density of Mk in N which defined as dk ¼ limn!1 ]f16i6n:i2M n the paper. Pm1 Suppose P ¼ ðpkj Þlm be a given matrix satisfying j¼0 pkj ¼ 1; pkj P 0 for k ¼ 1; 2; . . . ; l; j ¼ 0; 1; . . . ; m  1. For x 2 DN , let ðMk Þ

ðMk Þ

Nj

ðx; nÞ ¼ ]f1 6 i 6 n : i 2 M k ; xi ¼ jg; uk;j ðx; nÞ ¼

Nj

ðx; nÞ

ndk

;

then the modified Besicovitch subset of Moran fractal can be defined as:

Ks ¼ fwðxÞ : x 2 DN ; lim sup uk;j ðx; nÞ ¼ pkj ;

k ¼ 1; 2; . . . ; s;

j ¼ 0; 1; . . . ; m  1g;

n!1

where s is a positive integer less than l. And for the case s ¼ l, we denote by KðPÞ the modified Bisicovitch subset of Moran fractal. That is

KðPÞ ¼ fwðxÞ : x 2 DN ; lim uk;j ðx; nÞ ¼ pkj ; n!1

k ¼ 1; 2; . . . ; l;

j ¼ 0; 1; . . . ; m  1g:

In Section 3, we apply the vectorial multifractal analysis to get the Hausdorff dimension of the modified Besicovitch subset of Moran fractal, that is: Theorem 2.1. Let N k ðnÞ ¼ ]f1 6 i 6 n : i 2 Mk g, we have that

Ps dimH Ks ¼

k¼1 dk

Pm1

pkj log pkj  log mð1   Pm1 Pl j¼0 k¼1 dk pkj log r j

j¼0

Ps

k¼1 dk Þ

:

In Section 4, by using the traditional method, we get Hausdorff dimension results about the modified Besicovitch subset of Moran fractal. Note that Theorem 2.2 is the case for m=2: Theorem 2.2. Suppose 0 6 pk < 1; k 2 D, we set

(

KðpÞ ¼

wðxÞ : x 2 DN ; lim

n!1

) n 1X xmk ;j ¼ pk ; k 2 D ; n j¼1

then

Pl dk ðpk log pk þ ð1  pk Þ logð1  pk ÞÞ  dimH KðpÞ ¼ P k¼1 : Pl l d k¼1 k ð1  pk Þ log r 0 þ ð k¼1 dk pk Þ log r 1 As for the general case, we have:

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Theorem 2.3. Suppose P ¼ ðpkj Þlm be a given matrix satisfying have that

Pm1 j¼0

pkj ¼ 1; pkj P 0 for k ¼ 1; 2; . . . ; l;

j ¼ 0; 1; . . . ; m  1, we

Pl Pm1 k¼1 dk j¼0 pkj log pkj dimH KðPÞ ¼ Pm1 Pl : ð j¼0 k¼1 dk pkj Þ log r j Remark. We put 0 log 0 ¼ 0 all through this paper by convention. 3. Proof of Theorem 2.1 In this section, we will apply the vectorial multifractal analysis to get some dimension results about modified Besicovitch subset of Moran fractal. This also exhibits a good application of theorem in [21]. Let us briefly recall the notations and the main results in [21] first. Suppose ðX; dÞ be a metric space, Bðx; rÞ denote the ball fy 2 X : dðx; yÞ 6 rg, where x 2 X; r > 0. We assume that this metric space fulfills the Besicovitch covering property [3,4]. Given a function K from X  Rþ to E0 , the dual to a separable Banach space E. Several quantities on multifractal analysis as in [18] are defined as follows: For A # X; q 2 E; t 2 R; d > 0, we set   P 1 Pq;t ðAÞ ¼ sup j eðþt logðrj Þ Þ , where this supremum is taken on collections f xj ; rj g such that r j 6 d and  oK;d fB xj ; rj gj is a centered d-packing of A. q;t Pq;t K ðAÞ ¼ lim PK;d ðAÞ d!0 ( ) X q;t q;t PK ðAÞ ¼ inf PK ðF j Þ : A # [j F j j

Similarly, Hausdorff-like quantities are defined as follows:   P 1 Hq;t ðAÞ ¼ inf j eðhq;Kðxj ;rj Þiþt logðrj Þ Þ , where this infimum is taken over the families f xj ; r j g such that rj 6 d and  K;d fB xj ; rj gj is a centered d-cover of A. q;t Hq;t K ðAÞ ¼ lim HK;d ðAÞ; d!0

q;t Hq;t K ðAÞ ¼ sup HK ðAÞ: F #A

Accordingly, we have the dimensions:

Dimq ðAÞ ¼ infft 2 R : Pq;t K ðAÞ ¼ 0g;

BðqÞ ¼ Dimq ðXÞ;

dimq ðAÞ ¼ infft 2 R : Hq;t BðqÞ ¼ dimq ðXÞ; K ðAÞ ¼ 0g; Dq ðAÞ ¼ infft 2 R : Pq;t ðAÞ ¼ 0g; D ðqÞ ¼ Dq ðXÞ: K Of course, we also have some conclusions about dimensions as in [18]. For example, we have that the function BðqÞ is convex, and bðqÞ 6 BðqÞ 6 DðqÞ ¼ sðqÞ, where sðqÞ is the expansion of Minkovski–Bouligand dimension under multifractal circumstance. If a 2 E0 , and E # E, we set

Xða; EÞ ¼

  hx; Kðx; rÞi 6 hx; ai; for all x 2 E : x : lim sup  log r r!1

The set Xða; EÞ will simply be denoted by XðaÞ. This is the set of point x such that limr!1 Kðx;rÞ ¼ a.  log r If BðqÞ < 1, and v 2 E, one set

@ v BðaÞ ¼ lim t!0

Bðq þ tv Þ  BðqÞ t

B0 ðqÞ stands for the derivative of BðqÞ at point q when it exists, and @ v BðqÞ ¼ hv ; B0 ðqÞi. The main result in [21] is the following theorem: q;BðqÞ

Theorem 3.1 [21]. If for some q, the function BðqÞ is differentiable with derivative B0 ðqÞ and if HK bðqÞ ¼ BðqÞ and

dimXðB0 ðqÞÞ ¼ DimXðB0 ðqÞÞ ¼ B ðB0 ðqÞÞ where B ðaÞ ¼ inf q2E fha; qi þ BðqÞg is the Legendre transform of BðqÞ at point a.

ðAÞ > 0, then one has

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The following lemma is useful for checking the conditions of the above theorem.

l½q such that for l½q -almost every x one has

Lemma 3.2 [21]. If there exists a measure

lim sup r!0 q;BðqÞ

then HK

l½q ðBðx; rÞÞ

< 1;

1 eðhq;Kðx;rÞiþBðqÞ logðrÞ Þ

ðAÞ > 0.

We will use the above results to prove Theorem 2.1, and before proving Theorem 2.1, we make some preparations first: For some technical reasons, we put an assumption of gap condition here. Suppose that there exists d > 0 such that for any xn –x0n

distðJ x1 ;...;xn1 ;xn ; J x1 ;...;xn1 ;x0n Þ P dfjJ x1 ;...;xn1 ;xn j; jJx1 ;...;xn1 ;xn 0 jg:

ð3:1Þ

Without loss of generality, we can assume that 0 < d < 12, then we have the following lemma: Lemma 3.3. Suppose

m is a finite Borel measure supported on Ks , for any wðxÞ 2 Ks ; b > 0, we have

mðJx1 ;...;xn Þ

ab lim inf n!1

jJx1 ;...;xn j

mðBðwðxÞ; rÞÞ

6 lim inf

b

rb

r!0

6 lim sup

mðBðwðxÞ; rÞÞ

r!0

rb

6 ðadÞb lim sup n!1

mðJx1 ;...;xn Þ jJx1 ;...;xn jb

;

where a ¼ min06i6m1 fr i g. Proof of Lemma 3.3 For any wðxÞ 2 Ks ; r > 0, there exists n 2 N such that j J x1 ;...;xnþ1 j< r 6j J x1 ;...;xn j, together with

mðJx1 ;...;xnþ1 Þ 6 mðBðwðxÞ; j Jx1 ;...;xnþ1 jÞÞ, we have

mðBðwðxÞ; rÞÞ rb

P

mðBðwðxÞ; jJx1 ;...;xnþ1 jÞÞ jJ x1 ;...;xn jb

¼

jJ x1 ;...;xnþ1 jb

!

mðBðwðxÞ; jJx1 ;...;xnþ1 jÞÞ

jJ x1 ;...;xn jb

jJ x1 ;...;xnþ1 jb

P ðaÞb

mðBðwðxÞ; jJx1 ;...;xnþ1 jÞÞ jJx1 ;...;xnþ1 jb

:

Then it immediately follows from the above estimate that the left inequality holds. On the other hand, for any wðxÞ 2 Ks ; r > 0, there exists k 2 N such that d j J x1 ;...;xkþ1 j< r 6 d j J x1 ;...;xk j, with the gap condition (3.1), we assert that one of the two intervals ½wðxÞ  d j J x1 ;...;xk j; wðxÞ, ½wðxÞ; wðxÞ þ d j J x1 ;...;xk j is contained in J x1 ;...;xk , and mðJx1 ;...;xk Þ P mðBðwðxÞ; d j Jx1 ;...;xk jÞÞ, we have further

mðBðwðxÞ; rÞÞ rb

6

mðBðwðxÞ; djJx1 ;...;xk jÞÞ b

ðdjJ x1 ;...;xkþ1 jÞ

¼

which further implies the right inequality. Since the limit lim inf n!1 parable with

lðJx1 ;...;xn Þ jJx

1 ;...;xn

j

mðJx1 ;...;xn Þ jJx

1 ;...;xn

jb

jJ x1 ;...;xk jb djJ x1 ;...;xkþ1 j

!b

mðBðwðxÞ; djJx1 ;...;xk jÞÞ jJ x1 ;...;xk j

b

6 ðadÞb

mðBðwðxÞ; jJx1 ;...;xk jÞÞ jJ x1 ;...;xk jb



exists for b ¼ 1 in the proof of Theorem 2.1, we get from Lemma 3.3 that lðBðwðxÞrÞÞ is comr

, so we can use fJ x : x 2 D g instead of fBðxi ; r i Þg in the proof of Theorem 2.1.

For the simplicity, there is an assumption limn!1 ðndk  N k ðnÞÞ < 1 during the proof of Theorem 2.1. We note that Theorem 2.1 holds without such an assumption if we define a different cxj in the proof of Theorem 2.1. Proof of Theorem 2.1 For any wðxÞ 2 Ks , we define

lk ðJx1 ;...;xn i Þ ¼ lk ðJx1 ;...;xn Þ  then lk ðJ x1 ;...;xn Þ ¼ We set

Qm1 j¼0

ðMk Þ

Nj

pkj

ðx;nÞ



pkj ; if n þ 1 2 M k 1;

if n þ 1 R M k

for 1 6 k 6 s.

KðwðxÞ; jJx1 ;...;xn jÞ ¼  log lk ðJ x1 ;...;xn Þ16k6s ¼

m1 X

! ðM Þ Nj k ðx; nÞ log pkj

j¼0

16k6s

then we have

lim sup n!1

Let Sn ðqÞ ¼

P

KðwðxÞ; jJx1 ;...;xn jÞ ¼  log jJx1 ;...;xn j

jx1 ;...;xn j¼n

sðqÞ ¼ lim sup n!1

Pm1

 lim sup n!1

ðM k Þ ðx; nÞ log pkj j¼0 N j Pm1 Pl ðM Þ  j¼0 k¼1 Nj k ðx; nÞ log rj

!

Pm1

¼ 16k6s

lq11 ðJx1 ;...;xn Þlq22 ðJx1 ;...;xn Þ . . . lqs s ðJx1 ;...;xn Þ, then P P Pm1 qk log mð1  sk¼1 dk Þ þ sk¼1 dk log j¼0 pkj log Sn ðqÞ : ¼ Pm1 Pl  log jJx1 ;...;xn j ð d p Þ log r j j¼0 k¼1 k kj

j¼0 dk pkj log pkj Pm1 Pl j¼0 k¼1 dk pkj log r j

! ¼ b ðak Þ16k6s 16k6s

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Since

Ps

k¼1 dk qk

f ðq1 ; . . . ; qs Þ ¼< a; q > þsðqÞ ¼

Pm1 j¼0

P Pm1 qk P pkj log pkj  sk¼1 dk log j¼0 pkj  log mð1  sk¼1 dk Þ Pm1 Pl j¼0 ð k¼1 dk pkj Þ log r j

and for a 6 k 6 s,

@f ¼ @qk

dk

Xm1 j¼0

dk log

pkj log pkj þ

together with the fact that tion set. Hence

Pm1 j¼0

Pm1

qk j¼0 pkj log pkj Pm1 qk j¼0 pkj

!

.Xm1 Xl



 log r j :

pkj ¼ 1 holds for any a 6 k 6 s, we have qk ¼ 1 ð1 6 k 6 sÞ is the unique solution of the equa-

Ps

k¼1 dk

infs f ðq1 ; . . . ; qs Þ ¼ inf ha; qi þ sðqÞ ¼ f ð1; . . . ; 1Þ ¼

Pm1

pkj log pkj  log mð1  Pm1 Pl j¼0 ð k¼1 dk pkj Þ log r j

j¼0

q2E

q2R

dp k¼1 k kj

j¼0

Ps

k¼1 dk Þ

Now we would like to check the condition in Lemma 3.2. If q 2 Rs for 1 6 j 6 n, one sets

c xj ¼

8 q pkxk > > j >  d ; if j 2 Mk ; 1 6 k 6 s > s > < Q Pm1 pqk k kj

j¼0

k¼1

> > > > > : m1 ;

s S

if j R

Mk

k¼1

We define a measure

½q

l ðJx1 ;...;xn Þ ¼

l½q on DN as follows:

n Y

Qs Qm1 c xj ¼

k¼1

Qs

Pm1



1

k¼1 ð

j¼1

j¼0

q

j¼0

pkjk Þ

Pm1 j¼0

ðMk Þ

qk Nj

pkj

ðM k Þ

Nj

ðx;nÞ

ðx;nÞ

mm

Ps k¼1

N k ðnÞ

We have that

 

e


1 ;...;xn

jÞ>þsðqÞlog

1 jJ x ;...;xn j 1

Qs ¼ Qs

k¼1

k¼1

Qm1

P m1 j¼0

qk N

ðMk Þ

pkj j  n 

ðx;nÞ

j¼0 q

pkjk

dk

mnn

Ps

d k¼1 k



hence one has

!ndk P NðMk Þ ðx;nÞ m1



lim sup n!1

e

l½q ðJx1 ;...;xn Þ

 hq;Kðx;jJ x

1 ;...;xn

jÞþs

ðqÞlogjJ 1 ji x1 ;...;xn

 ¼ lim sup n!1

¼ lim sup n!1

s m1 Y X k¼1 s Y k¼1

s P

j

j¼0

q pkjk

m

k¼1

Nk ðnÞn

s P

dk

k¼1

j¼0

Pm1 j¼0

q

pkjk

!ndk Nk ðnÞ <1

m

Therefore by Theorem 3.1 and Lemma 3.2 we have

Ps dimXðs0 ðqÞÞ ¼ infs f< a; q > þsðqÞg ¼ f ð1; . . . ; 1Þ ¼ q2R

k¼1 dk

Pm1

pkj log pkj  log mð1  Pm1 Pl j¼0 ð k¼1 dk pkj Þ log r j

j¼0

Ps

k¼1 dk Þ

Since the set Xðs0 ðqÞÞ is the same set with Ks . h 4. Proof of Theorem 2.2 and Theorem 2.3 In this section, we will use traditional method to consider the modified Besicovitch subset of Moran fractal. Obviously Theorem 2.2 is a special case of Theorem 2.3, so we only prove Theorem 2.3. Before proving Theorem 2.3, let us look at a useful lemma first: Lemma 4.1 [22]. Let fX n : n P 1g be a sequence of independent random variables with finite second moments. If there are Pn Pn P X  EðX k Þ VðX n Þ k¼1 k k¼1 < 1, then lim ¼ 0: a:e. positive numbers an such that an % 1 and 1 n!1 2 n¼1 a an n

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Proof of Theorem 2.3 Let

log lðJ x1 x2 ...xn Þ ¼

n X

lðJx1 x2 ...xn Þ ¼ tx1 tx2 . . . txn , where txi ¼ pkj if i 2 Mk and xi ¼ j, then log txi ¼

l X X

log t xi :

k¼1 16i6n i2Mk

i¼1

Obviously we have

X

log t xi ¼

16i6n i2M k

m1 X j¼0

]f1 6 i 6 n : i 2 M k ; xi ¼ jg log pkj ¼ b

m1 X

ðMk Þ

Nj

ðx; nÞ log pkj

j¼0

and ðMk Þ

Nj

ðx; nÞ

n

! dk pkj as n ! 1:

ð4:1Þ

If we denote by

Nj ðx; nÞ ¼ ]f1 6 i 6 n : xi ¼ jg then l X

Nj ðx; nÞ ¼

ðMk Þ

Nj

ðx; nÞ:

ð4:2Þ

k¼1

Together with (4.1 and 4.2), we have that

log jJ x1 x2 ...xn j ¼ log rx1 r x2 . . . r xn ¼

m1 X

Nj ðx; nÞ log r j ¼

j¼0

m1 X

l X

ðMk Þ

Nj

ðx; nÞ log rj

j¼0 k¼1

With the above equalities, we have further

lim

n!1

Pl Pm1 ðMk Þ Pl Pm1 ðx; nÞ log pkj log lðJ x1 x2 ...xn Þ k¼1 j¼0 N j k¼1 dk j¼0 pkj log pkj ¼ ¼ lim Pm1 Pl P P ðM k Þ m1 l n!1 log jJ x1 x2 ...xn j ðx; nÞ log rj j¼0 k¼1 N j¼0 ð k¼1 dk pkj log pkj Þ log r j j

fwðxmk;j Þg1 j¼1

Let be random variables with independent and identically distribution, it follows from Lemma 4.1 that lðKðPÞÞ ¼ 1. Then by Billingsley’s theorem [6], we have

Pl

dimH KðPÞ ¼

Pm1 k¼1 dk j¼0 pkj log pkj Pm1 Pl j¼0 ð k¼1 dk pkj Þ log r j

:



Remark 1. If we do not partition N, that is if l ¼ 1, then we have that

(

KðPÞ ¼

wðxÞ : lim

n!1

) n 1X 1j ðxk Þ ¼ pj ; j 2 D ; n k¼1

Pm1 pj log pj and the Hausdorff dimension of KðPÞ is Pj¼0 , which coincides with the dimension result in [11]. m1 j¼0

pj log r j

2. It deserves to point out that EI Naschie [13–17] have achieved many valuable results on the relevant fields for example applications in quantum mechanics and E-infinity theory, our work is relevant in physics for relation between the dimension theory and high energy physics. Therefore researches concerning fractals and multifractals are very significant for physics [20].

References [1] [2] [3] [4] [5]

Barreira L, Saussol B, Schmeling J. Distribution of frequencies of digits via multifractal analysis. J Num Theory 2002;97:410–38. Besicovitch AS. On the sum of digits of real numbers represented in the dyadic system. Math Ann 1934;110:321–30. Besicovitch AS. A general form of covering principle and relative differentiation of additive function. Pro Cambridge Philos Soc 1945;41:103–10. Besicovitch AS. A general form of covering principle and relative differentiation of additive function II. Pro Cambridge Philos Soc 1946;42:1–10. Bisbas A, Karanikas C, Proios G. On the distribution of digits in dyadic expansions. Result Math 1998;33:40–9.

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