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Sections of Sets of zero Lebesgue measure

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

K. J. Falconer
K. J. Falconer
Affiliation:
Corpus Christi College, Cambridge, CB2 1RH.
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Extract

§1. Introduction and notation. In [1] and [2], Besicovitch demonstrated that there exist plane sets of measure zero containing line segments (and indeed entire lines) in all directions in the plane. It is natural to ask about the existence of analogous sets in Euclidean spaces of higher dimensions, and in [3] we defined an (n, k)-Besicovitch set to be a subset A of Rn, of n-dimensional Lebesgue measure zero, such that for each k-dimensional subspace Π of Rn, some translate of Π intersects A in a set of positive k-dimensional measure. (Thus Besicovitch's original constructions were for (2,1)-Besicovitch sets.) Recently, Marstrand [5] has shown (by approximating to sets by unions of cubes) that no (3, 2)-Besicovitch sets exist, and simultaneously the author [3] proved using Fourier transform methods that (n, k)- Bsicovitch sets cannot exist if k > ½n.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 90 - 96
Copyright
Copyright © University College London 1980

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References

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