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Packing dimension, Hausdorff dimension and Cartesian product sets

Published online by Cambridge University Press:  24 October 2008

Yimin Xiao
Yimin Xiao
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, U.S.A e-mail: xiao@math.ohio-state.edu
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Abstract

We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 3 , October 1996 , pp. 535 - 546
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Besicovitch, A. S. and Moran, P. A. P.. The measure of product and cylinder sets. J. London Math. Soc. 20 (1945), 110–120.Google Scholar
[2]Eggleston, H. G.. A correction to a paper on the dimension of Cartesian product sets. Proc. Camb. Phil. Soc. 49 (1953), 437–440.CrossRefGoogle Scholar
[3]Falconer, K. J.. Fractal geometry - mathematical foundations and applications (Wiley & Sons, 1990).CrossRefGoogle Scholar
[4]Hasse, H.. Non-σ-finite sets for packing measure. Mathematika 33 (1986), 129–136.CrossRefGoogle Scholar
[5]Hasse, H.. On the dimension of product measures. Mathematika 37 (1990), 316–323.CrossRefGoogle Scholar
[6]Howroyd, J. D.. On Hausdorff and packing dimension of product spaces. Math. Proc. Camb. Phil Soc. 119 (1996), 715–727.CrossRefGoogle Scholar
[7]Hu, X. and Taylor, S. J.. Fractal properties of products and projections of measures in Rd. Math. Proc. Camb. Phil. Soc. 115 (1994), 527–544.CrossRefGoogle Scholar
[8]Joyce, H. and Preiss, D.. On the existence of subsets of finite positive packing measure. Mathematica 42 (1995), 15–24.Google Scholar
[9]Kaufman, R.. Entropy, dimension and random sets. Quart. J. Math. Oxford (2) 38 (1987), 77–80.CrossRefGoogle Scholar
[10]Marstrand, J. M.. The dimension of the Cartesian product sets. Proc. Camb. Phil. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
[11]Rogers, C. A.. Hausdorff measures (Cambridge University Press, 1970).Google Scholar
[12]Rogers, C. A. and Taylor, S. J.. Additive set functions in euclidean space. Ada Math. 101 (1959), 273–302.Google Scholar
[13]Raymond, X. Saint and Tricot, C.. Packing regularity of sets in n–space. Math. Proc. Camb. Phil. Soc. 103 (1988), 133–145.CrossRefGoogle Scholar
[14]Taylor, S. J.. The measure theory of random fractals. Math. Proc. Camb. Phil. Soc. 100 (1986), 383–406.CrossRefGoogle Scholar
[15]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar
[16]Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.CrossRefGoogle Scholar
[17]Taylor, S. J. and Tricot, C.. The packing measure of rectifiable subsets of the plane. Math. Proc. Phil. Soc. 99 (1986), 285–296.CrossRefGoogle Scholar
[18]Talagrand, M. and Xiao, Yimin. Fractional Brownian motion and packing dimension. J. Theoret. Probab., to appear.Google Scholar