On a planar variant of the Kakeya problem

A {\it $\mathcal{K}^n_2$-set} is a set of zero Lebesgue measure containing a translate of every plane in an $(n-2)$–dimensional manifold in $\mathrm{Gr}(n,2)$, 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 $\mathcal{K}^4_2$-sets~$E$. When the underlying field is replaced by $\C$, we get $\dim_H(E)\ge 7$ for all $\mathcal{K}^4_2$-sets over $\C$, and we construct an example to show that this is sharp. Thus $\mathcal{K}^4_2$-sets over $\C$ do not necessarily have full Hausdorff dimension.

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Published 1 January 2006