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Local dimension and regular points

Published online by Cambridge University Press:  24 October 2008

Martin T. Barlow  and
S. James Taylor
Martin T. Barlow
Affiliation:
Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB
S. James Taylor
Affiliation:
Department of Mathematics, Maths-Astronomy Building, University of Virginia, Charlottesville, VA 22903, U.S.A.
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Extract

The Hausdorff–Besicovitch dimension of a set A ⊆ ℝd, denoted dim (A), relates to the structure of A in the neighbourhood of its thickest point. If A is irregular then one may wish for an index which will describe the size of A in different places. The obvious definition of the local dimension of A near x is

and it is not hard to verify that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 361 - 368
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Barlow, M. T. and Taylor, S. J.. Defining fractal subsets of ℤd. Proc. London Math. Soc. (3) 64 (1992), 125–152.CrossRefGoogle Scholar
[2]Davies, R. O.. A property of Hausdorff measures. Proc. Cambridge Philos. Soc. 52 (1956), 30–32.CrossRefGoogle Scholar
[3]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[4]Frostman, O.. Potential d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Medd. Lunds. Math. Sem. 3 (1935), 1–118.Google Scholar
[5]Landhof, N. S.. Foundations of Modern Potential Theory (Springer-Verlag, 1972).CrossRefGoogle Scholar
[6]Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar
[7]Rogers, C. A. and Taylor, S. J.. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961), 1–31.CrossRefGoogle Scholar
[8]Rogers, C. A.. Sets non-σ-finite for Hausdorff measure. Mathematika 9 (1962), 95–103.CrossRefGoogle Scholar
[9]Sion, M. and Sjerve, D.. Approximation properties of measures generated by continuous set functions. Mathematika 9 (1962) 145–156.CrossRefGoogle Scholar
[10]Taylor, S. J.. On the connection between Hausdorff measures and generalised capacities. Proc. Cambridge Philos. Soc. 57 (1961), 524–531.CrossRefGoogle Scholar