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Projection theorems for box and packing dimensions

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer  and
J. D. Howroyd
K. J. Falconer
Affiliation:
Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
J. D. Howroyd
Affiliation:
Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
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Abstract

We show that if E is an analytic subset of ℝn then

for almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 2 , February 1996 , pp. 287 - 295
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Falconer, K. J.. Fractal geometry - mathematical foundations and applications (John Wiley, 1990).CrossRefGoogle Scholar
[2]Falconer, K. J.. Hausdorff dimension and the exceptional set of projections. Mathematika 29 (1982), 109–115.CrossRefGoogle Scholar
[3]Haase, H.. Non-σ-nnite sets for packing measure. Mathematika 33 (1986), 129–136.CrossRefGoogle Scholar
[4]Hu, X. and Taylor, S. J.. Fractal properties of products and projections of measures in ℝd. Math. Proc. Cambridge Philos. Soc. 115 (1994), 527–544.CrossRefGoogle Scholar
[5]Järvenpää, M.. On the upper Minkowski dimension, the packing dimension, and orthogonal projections. Annales Acad. Sci. Fen. A Dissertationes 99 (1994).Google Scholar
[6]Kaufman, R.. On Hausdorff dimension of projections. Mathematika 15 (1968), 153–155.CrossRefGoogle Scholar
[7]Marstrand, J. M.. Some fundamental properties of plane sets of fractional dimension. Proc. London Math. Soc. (3) 4 (1954), 257–302.CrossRefGoogle Scholar
[8]Mattila, P.. Hausdorff dimension, orthogonal projections and intersections with planes. Annales Acad. Sci. Fen. A 1 (1975), 227–244.Google Scholar
[9]Mattila, P.. Geometry of sets and measures in Euclidean spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar
[10]Tricot, C.. Rarefaction indices. Mathematika 27 (1980), 46–57.CrossRefGoogle Scholar
[11]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar