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Classes of sets with large intersection

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

K. J. Falconer
K. J. Falconer
Affiliation:
School of Mathematics, University Walk, Bristol, BS8 1TW
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Extract

If E1 and E2 are subsets of ℝn and a- is an isometry or similarity transformation, it is useful to be able to estimate the Hausdorff dimension of E1 ∩ σ(E2) in terms of the dimensions of E1 and E2. If E1 and E2 are compact, then, as σvaries, dim (El ∩ σ(E2)) is “in general” at most max (dim E1 + dim E2 − n, 0) and “often” at least this value (see Mattila [9] and Kahane [7] for more precise statements of these ideas). However, as we shall see, it is possible to construct non-compact sets E of any given dimension that are “sufficiently dense” in ℝn to ensure that dim (E ∩ σ(E)) = dim E for all similarities σ More generally, we shall show that for each s there are large classes of sets & of dimensions between s and n, closed under reasonable transformations including similarities, such that the intersection of any countable collection of sets in & has dimension at least s. Such collections of sets are required, for example, in the constructions of subsets of ℝn with certain dimensional properties given by Davies [1] and Falconer [5].

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 32 , Issue 2 , December 1985 , pp. 191 - 205
Copyright
Copyright © University College London 1985

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References

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