On a planar variant of the Kakeya problem
Classical Analysis and ODEs
2007-05-23 v2
Abstract
A K^n_2-set is a set of zero Lebesgue measure containing a translate of every plane in an (n-2)-dimensional manifold in Gr(n,2), where the manifold fulfills a curvature condition. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dim_H(E)\ge 7/2 for all K^4_2-sets E. When the underlying field is replaced by the complex numbers C, we get dim_H(E)\ge 7 for all K^4_2-sets over C, and we construct an example to show that this is sharp. Thus K^4_2-sets over C do not necessarily have full Hausdorff dimension.
Keywords
Cite
@article{arxiv.math/0502360,
title = {On a planar variant of the Kakeya problem},
author = {Keith M. Rogers},
journal= {arXiv preprint arXiv:math/0502360},
year = {2007}
}
Comments
14 pages, to appear, Math. Res. Lett. Includes a mathematical correction, and changes to the acknowledgements