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

TL;DR

This paper constructs Kakeya-type sets with directions from Cantor sets on smooth curves in higher dimensions, showing such sets have small measure and lead to unbounded directional maximal operators.

Contribution

It extends planar Kakeya set constructions to higher dimensions using probabilistic methods based on the tree structure of Cantor sets of directions.

Findings

Existence of small measure Kakeya-type sets with Cantor set directions

Unboundedness of the associated directional maximal operator on L^p spaces

Extension of planar results to higher dimensions

Abstract

Given a Cantor-type subset $\Omega$ of a smooth curve in $\mathbb R^{d+1}$, we construct examples of sets that contain unit line segments with directions from $\Omega$ 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 $\Omega$, 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 $\Omega$ is unbounded on $L^p(\mathbb{R}^{d+1})$ for all $1\leq p<\infty$.