Kakeya-type sets over Cantor sets of directions in $\mathbb{R}^{d+1}$

Edward Kroc; Malabika Pramanik

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arXiv · 1404.6235

Kakeya-type sets over Cantor sets of directions in $\mathbb{R}^{d+1}$

Edward Kroc·Malabika Pramanik

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Abstract

Given a Cantor-type subset $Ω$ of a smooth curve in $\mathbb R^{d+1}$, we construct examples of sets that contain unit line segments with directions from $Ω$ and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small $(d+1)$-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of $Ω$, and extends to higher dimensions an analogous planar result of Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set $Ω$ is unbounded on $L^p(\mathbb{R}^{d+1})$ for all $1\leq p<\infty$.

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BibTeX RIS

Edward Kroc, Malabika Pramanik. 2014-04-24. Kakeya-type sets over Cantor sets of directions in $\mathbb{R}^{d+1}$. https://arxiv.org/abs/1404.6235

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