L. D. NIEM ET AL.275

1) Can one find the precise scaling function

such

that ? Is the set

0< <B

0,11\ B0,

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

R is

immeasurable.

2) What is the relationship between

x and

2

x?

3) How large is the set of points

such that

=?

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

-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

-Adic Expan-

sion,” Mathematical Proceedings of the Cambridge Phi-

losophical Society, Vol. 136, No. 1, 2004, pp. 139-165.

doi:10.1017/S0305004103007047

Copyright © 2011 SciRes. APM