Kakeya Sets over the Heisenberg Group

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 H^1. By utilizing the sub-Riemannian manifold of H^1 we can apply tools in geometric measure theory which at this time cannot be applied in R^3. 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 R^3.