On a Planar Variant of the Kakeya Problem

An (n − 2)–dimensional smooth manifold in Gr(n, 2) is said to be curved if its restriction to each m–space of R is no more than (m − 2)– dimensional. A K 2 -set is a set of zero Lebesgue measure containing a translate of every plane in a curved manifold. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dimH(E) ≥ 7/2 for all K 2 -sets E. When the underlying field is replaced by C, we get dimH(E) ≥ 7 for all K 2 -sets over C, and we construct an example to show that this is sharp. Thus K 2 -sets over C do not necessarily have full Hausdorff dimension.