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Journal of the London Mathematical Society 1994 49(2):267-280; doi:10.1112/jlms/49.2.267
© 1994 by London Mathematical Society
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© The London Mathematical Society

Sets with Large Intersection Properties

K. J. Falconer

School of Mathematics, University Walk Bristol BS8 1TW and Mathematical Institute, University of St Andrews North Haugh, St Andrews, Fife KYI6 9SS

Received 4 June 1992.

For 0 < s ≤ n let Gs be the class of G{delta}-subsets of Rn such that F isin Gs if Formula has Hausdorff dimension at least s for all sequences of similarity transformations Formula. We show that Gs is closed under countable intersections and under bi-Lipschitz functions, and thus is the maximal class of G{delta}-sets of Hausdorff dimension at least s that is closed under countable intersection and similarities. We also show that sets in Gs must have packing dimension n. Many examples of Gs-sets occur in Diophantine approximation.


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