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Kakeya Sets in Cantor directions

Michael D. Bateman·Nets Hawk Katz
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Abstract

We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union of the doubles has area log log N/ log N. In particular, this implies unbounded of the associated maximal operator in L^p for any p different from infinity. The construction is by randomizing an earlier construction of the second author for the L^2 case. The proof that the construction satisfies the desired conditions is by elementary estimates in the theory of percolation on trees as developed by R. Lyons.

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Michael D. Bateman, Nets Hawk Katz. 2006-09-06. Kakeya Sets in Cantor directions. https://arxiv.org/abs/math/0609187

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