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Lebesgue density influences Hausdorff measure; large sets surface-like from many directions

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Roy O. Davies  and
Henryk Fast
Roy O. Davies
Affiliation:
Department of Mathematics, The University, Leicester, LE1 7RH, England.
Henryk Fast
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.
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Extract

In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets Fn, whose intersection has some required property; it is desired that ∩ Fn shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each Fn+1 sufficiently “dense” in Fn. Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝk have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 25 , Issue 1 , June 1978 , pp. 116 - 119
Copyright
Copyright © University College London 1978

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References

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