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Kakeya-Type Sets Over Cantor Sets of Directions in \(\mathbb {R}^{d+1}\)

  • Edward Kroc
  • Malabika PramanikEmail author
Article
  • 139 Downloads

Abstract

Given a Cantor-type subset \(\Omega \) of a smooth curve in \(\mathbb R^{d+1}\), we construct examples of sets that contain unit line segments with directions from \(\Omega \) and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small \((d+1)\)-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of \(\Omega \), and extends to higher dimensions an analogous planar result of Bateman and Katz (Math Res Lett 15(1):73–81, 2008). In contrast to the planar situation, a significant aspect of our analysis is the classification of intersecting tube tuples relative to their location, and the deduction of intersection probabilities of such tubes generated by a random mechanism. The existence of these Kakeya-type sets implies that the directional maximal operator associated with the direction set \(\Omega \) is unbounded on \(L^p(\mathbb {R}^{d+1})\) for all \(1\le p<\infty \).

Keywords

Kakeya sets Directional maximal operators Cantor sets 

Mathematics Subject Classification

Primary 28A75 42B25 Secondary 60K35 

Notes

Acknowledgments

We would like to thank the two anonymous referees for useful comments and a careful reading of the manuscript. The second author would like to thank Gordon Slade of the Department of Mathematics at the University of British Columbia for a helpful discussion on percolation theory. The research was partially supported by an NSERC Discovery Grant.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada

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