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On a planar variant of the Kakeya problem

Keith M. Rogers
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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.

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Keith M. Rogers. 2005-11-29. On a planar variant of the Kakeya problem. https://arxiv.org/abs/math/0502360

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