Advances in Pure Mathematics, 2011, 1, 274-275
doi:10.4236/apm.2011.15048 Published Online September 2011 (http://www.SciRP.org/journal/apm)
Besicovitch-Eggleston Function
Manav Das
Department of Mat hematics, University of Louisville, Louisville, USA
Received June 2, 2011; revised July 2, 2011; accepted July 15, 20 11
Abstract
In this work we introduce a function based on the well-known Besicovitch-Eggleston sets, and prove that the
Hausdorff dimension of its graph is 2.
Keywords: Hausdorff Dimension, Multifractal, Binomial Measure, Dyadic Intervals
1. Introduction
Let


0,1x, and let denote its
binary expansion. For any 12
=.,, =0 or 1
i
xxx x
0< <12p we may consider
the set

12
=0,1: limsup n
pn
xx x
K
x
n





p



Besicovitch [2] prov ed that

 
log1log 1
dim =log2
Hp
pp pp
K 
where dimH
A
denotes the Hausdorff dimension of the
set
A
. This result was generalized to the -ary case
by Eggleston [8]. Billing sley proved a more general ver-
sion of this result in the context of probability spaces [3].
Billingsley’s result was related to densities in [5], and a
similar result involving packing dimensions was proved
in [6]. Sets such as
N
p
K
are studied in the context of
multifractal theory (see [1,7,9,11,14-16]) and Billing-
sley-type results have been proved by several authors in
this context. Recently, such a result has been proved for
a countable symbol space in [13].
In this paper, we are interested in a natural function
that may be defined using Besicovitch’s result. We call
this the Besicovitch-Eggleston function: let


1,,= #Nnx
of ’s in the first digits of the dyadic expansion for 1n
x
. Define



1, ,ifit exists
lim
=
1othe
n
Nnx
fx n





rwise
This function allows us to visualize the multifractal
components of


0,1 as lev el sets. If we let
p

denote
the invariant multifractal meas ure on
p
K
then it is clear
that




d=
p.
f
yy 

>0
p
In the next section, we state and prove our main result,
and finally we close with some open problems.
2. Main Result
We will need the following result by Besicovitch and
Moran. This is not the form in which it was originally
stated and proved. However, this modern version may be
found in [12].
Theorem 1. For any there exists a constant
s such that for all Borel sets we must
have

0,1s
b2 ER


1

d
ss
sy
Eb Ey 
where




E=:,
y
Exxy.
We are now ready to state the main result:
Theorem 2. Let
 



=0,1Graph fB
= 2
HB0,1. Then
and
dim


2B
p
=0.
Proof. The upper bound is obvious while the lower
bound follows from Theorem 1. Fix and

0,1s
choose such that log logqq >.
log 2
pp

s
Therefore





1=fp s

. We can choose an interval
s
I
containing such that for every
p

=
sy
B
s
yI

.
It follows from Theorem 1 that . Observ-
ing that was arbitrary gives us the lower bound.
Moreover, since every vertical line meets exactly
once, Fubini’s theorem tells us that in fact

=B
B
1s

<1s


=0
2
B.
3. Concluding Remarks
Here we pose some problems related to the Besicovitch-
Eggleston function.

L. D. NIEM ET AL.275
1) Can one find the precise scaling function

such
that ? Is the set


0< <B





0,11\ B0,
immesaurable, that is, either null or non-

-finite for
every translation invariant Borel measure on ? See
[10], where it is shown that the set
2
R
<1/20<
\
p
p
K
R is
immeasurable.
2) What is the relationship between


f
x and

2
f
x?
3) How large is the set of points
x
such that

=?
x
fx Can we characterize this set of fixed points?
4. References
[1] L. Barreira, B. Saussol and J. Schmeling, “Distribution of
Frequencies of Digits via Multifractal Analysis,” Journal
of Number Theory, Vol. 97, No. 2, 2002, pp. 410-438.
doi:10.1016/S0022-314X(02)00003-3
[2] A. Besicovitch, “On the Sum of Digits of Real Numbers
Represented in the Dyadic System,” Mathematische An-
nalen, Vol. 110, No. 1, 1934, pp. 321-330.
doi:10.1007/BF01448030
[3] P. Billingsley, “Hausdorff Dimension in Probability The-
ory II,” Illinois Journal of Mathematics, Vol. 5, No. 2,
1961, pp. 291-298.
[4] H. Cajar, “Billingsley Dimension in Probability Spaces,”
Springer-Verlag, Berlin-New York, 1981.
[5] C. S. Dai and S. J. Taylor, “Defining Fractals in a Prob-
ability Space,” Illinois Journal of Mathematics, Vol. 38,
No. 3 1994, pp. 480-500.
[6] M. Das, “Billingsley’s Packing Dimension,” Proceedings
of the American Mathematical Society, Vol. 136, No. 1,
2008, pp. 273-278. doi:10.1090/S0002-9939-07-09069-7
[7] M. Das, “Hausdorff Measures, Dimensions and Mutual
Singularity,” Transactions of the American Mathematical
Society, Vol. 357, No. 11, 2005, pp. 4249-4268.
doi:10.1090/S0002-9947-05-04031-6
[8] H. G. Eggleston, “The Fractional Dimension of a Set
Defined by Decimal Properties,” Quarterly Journal of
Mathematics—Oxford Journals, Vol. 2, No. 20, 1949, pp.
31-36.
[9] G. A. Edgar, “Measure, Topology, and Fractal Geome-
try,” Springer-Verlag, New York, 1990.
[10] M. Elekes and T. Keleti, “Borel Sets which are Null or
Non-

-Finite for Every Translation Invariant Measure,”
Advances in Mathematics, Vol. 201, No. 1, 2006, pp.
102-115. doi:10.1016/j.aim.2004.11.009
[11] K. J. Falconer, “Techniques in Fractal Geometry,” John
Wiley & Sons, Ltd., Chichester, 1997.
[12] K. J. Falconer, “The Geometry of Fractal Sets,” Cam-
bridge University Press, Cambridge, 1986.
[13] A. H. Fan, L. M. Liao, J. H. Ma and B. W. Wang, “Di-
mension of Besicovitch-Eggleston Sets in Countable
Symbolic Space,” Nonlinearity, Vol. 23, No. 5, 2010, pp.
1185-1197.
[14] L. Olsen, “On the Hausdorff Dimension of Generalized
Besicovitch-Eggleston Sets of d-Tuples of Numbers,”
Indagationes Mathematicae, Vol. 15, No. 4, 2004, pp.
535-547. doi:10.1016/S0019-3577(04)80017-X
[15] L. Olsen, “Applications of multifractal divergence points
to some sets of d-tuples of numbers defined by their
N
-adic expansion,” Bulletin des Sciences Mathé-
matiques, Vol. 128, No. 4, 2004, pp. 265-289.
[16] L. Olsen, “Applications of Multifractal Divergence Points
to Sets of Numbers Defined by Their
N
-Adic Expan-
sion,” Mathematical Proceedings of the Cambridge Phi-
losophical Society, Vol. 136, No. 1, 2004, pp. 139-165.
doi:10.1017/S0305004103007047