March 3, 2026

Kakeya Sets over the Heisenberg Group

Key Points

Kakeya sets can be sharply bounded in the Heisenberg group, presenting a unique geometric challenge and opportunity.
Sharp bounds found by Jiayin Liu reveal new insights into the hausdorff dimension of these sets in non-Euclidean spaces.
Construction of Kakeya sets utilizes the sub-riemannian manifold of the Heisenberg group, enhancing geometric measure theory applications.
Significant findings expand the understanding of Kakeya sets beyond traditional Euclidean spaces, indicating broader implications.

Abstract

The Hausdorff dimension of Kakeya sets is an interesting problem where the two-dimensional case can be proven directly, but even obtaining bounds on the higher dimension analogues can require highly technical machinery. The difficulty of the general case has inspired analysts to look at Kakeya sets from non-Euclidean viewpoints. In this paper, we explore a construction of the Kakeya set in the first Heisenberg group H¹. By utilizing the sub-Riemannian manifold of H¹ we can apply tools in geometric measure theory which at this time cannot be applied in R³. We restate and provide a detailed proof for a sharp bound obtained by Jiayin Liu for these so-called Kakeya-Heisenberg sets. We also discuss recent results in other non-Euclidean spaces and in R³.

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Kakeya Sets over the Heisenberg Group

Key Points

Abstract

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