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Order-two density of sets and measures with non-integral dimension

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

K. J. Falconer  and
O. B. Springer
K. J. Falconer
Affiliation:
The Mathematical Institute, The University of St. Andrews, North Haugh, St. Andrews, Fife, KYI6 9SS, Scotland.
O. B. Springer
Affiliation:
Credit Suiss Financial Products, 1 Cabot Square, London, E14 4QJ.
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Extract

This paper is concerned with the geometry of a measure μ, and in particular with the relationship between various .s-dimensional densities of μ, the geometry of the support of μ and the question of whether s is an integer.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 1 - 14
Copyright
Copyright © University College London 1995

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References

1.Bedford, T. and Fisher, A. M.. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. (3), 64 (1992), 95–124.CrossRefGoogle Scholar
2.Besicovitch, A. S.. On the fundamental geometrical properties of linearly measurable sets of points. Math. Annalen, 98 (1928), 422–464.CrossRefGoogle Scholar
3.Besicovitch, A. S.. On linear sets of points of fractional dimension. Math. Annalen, 101 (1929), 161–193.CrossRefGoogle Scholar
4.Besicovitch, A. S.. On the fundmental geometrical properties of linearly measurable sets of points II. Math. Annalen, 115 (1938), 296–329.CrossRefGoogle Scholar
5.Falconer, K. J.. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics 85 (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
6.Falconer, K. J.. Wavelet transforms and order-two densities of fractals. J. Statistical Physics, 67 (1992), 781–793.CrossRefGoogle Scholar
7.Federer, H.. Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften 153 (Springer-Verlag, Berlin-Heidelberg-New York, 1969)Google Scholar
8.Marstrand, J. M.. Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London Math. Soc. (3), 4 (1954), 257–302.CrossRefGoogle Scholar
9.Marstrand, J. M.. Circular density of plane sets. J. London Math. Soc. 30 (1955), 238–246.CrossRefGoogle Scholar
10.Marstrand, J. M.. Hausdorff two-dimensional measure in 3-space. Proc. London Math. Soc. (3), 11 (1961), 91–108.CrossRefGoogle Scholar
11.Marstrand, J. M.. The (φ, n)-regular subsets of n-space. Trans. American Math. Soc. 113 (1964), 369–392.Google Scholar
12.Mattila, P.. Hausdorff m regular and rectifiable sets in n-space. Trans. American Math. Soc. 205 (1975), 263–274.Google Scholar
13.Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces. (Cambridge University Press, 1995).CrossRefGoogle Scholar
14.Moore, E. F.. Density ratios and (φ, 1) rectifiability in n-space. Trans. American Math. Soc. 69 (1950), 324–334.Google Scholar
15.O'Neil, T. C.. A local version of the projection theorem and other results in geometric measure theory. Ph.D. Thesis (University of London, 1994).Google Scholar
16.Preiss, D.. Geometry of measures in Rn: Distribution, rectifiability and densities. Annals of Math. 125 (1987), 537–643.CrossRefGoogle Scholar
17.Springer, O. B.. Order-two Density and Self-conformal Sets. PhD thesis (University of Bristol, 1993).Google Scholar