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.
References
[1]
L. Barreira, B. Saussol and J. Schmeling, “Distribution of Fre-quencies 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 Annalen, Vol. 110, No. 1, 1934, pp. 321-330.
doi:10.1007/BF01448030
[3]
P. Billingsley, “Hausdorff Di-mension in Probability Theory 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 Probability 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 Prop-erties,” Quarterly Journal of Mathematics—Oxford Journals, Vol. 2, No. 20, 1949, pp. 31-36.
[9]
G. A. Edgar, “Measure, Topology, and Fractal Geometry,” 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, “Tech-niques in Fractal Geometry,” John Wiley & Sons, Ltd., Chichester, 1997.
[12]
K. J. Falconer, “The Geometry of Fractal Sets,” Cambridge University Press, Cambridge, 1986.
[13]
A. H. Fan, L. M. Liao, J. H. Ma and B. W. Wang, “Dimension 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 -Tuples of Numbers,” Indagationes Mathematicae, Vol. 15, No. 4, 2004, pp. 535-547. doi:10.1016/S0019-3577(04)80017-X
[15]
L. Olsen, “Appli-cations of multifractal divergence points to some sets of -tuples of numbers defined by their -adic expansion,” Bulletin des Sciences Mathématiques, Vol. 128, No. 4, 2004, pp. 265-289.
[16]
L. Olsen, “Applications of Multifractal Diver-gence Points to Sets of Numbers Defined by Their -Adic Expansion,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 136, No. 1, 2004, pp. 139-165.
doi:10.1017/S0305004103007047
