A non-sticky Kakeya set of Lebesgue measure zero

TL;DR

This paper constructs explicit examples of measure-zero Kakeya sets that are non-sticky in two and higher dimensions, advancing understanding of their geometric properties relevant to the Kakeya conjecture.

Contribution

It provides explicit constructions of non-sticky Lebesgue measure zero Kakeya sets in any dimension, including novel examples not formed by simple Cartesian products.

Findings

Constructed a measure-zero non-sticky Kakeya set in ${ m R}^2$.

Built high-dimensional Kakeya sets that are non-trivial, non-sticky, and have Hausdorff dimension d.

Abstract

The Kakeya set conjecture in ${\mathbb R} ^3$ was recently resolved by Wang and Zahl. The distinction between sticky and non-sticky Kakeya sets plays an important role in their proof. Although the proof did not require the Kakeya set to be Lebesgue measure zero, measure zero Kakeya sets are the crucial case whose study is required to resolve the conjecture. In this paper, we explicitly construct a non-sticky Kakeya set of Lebesgue measure zero in ${\mathbb R}^2$ (and hence in any dimension). We also construct non-trivial sticky and non-sticky Kakeya sets in high dimension that are not formed by taking the Cartesian product of a 2-dimensional Kakeya set with ${\mathbb R}^{d-2}$, and we verify that both Kakeya sets have Hausdorff dimension $d$.