On the Dimension of the CC-geodesic Kakeya sets in the first Heisenberg group

We study CC-geodesic Kakeya sets in the first Heisenberg group, defined by requiring a left translate of every unit-speed CC-geodesic segment issuing from the identity.

The natural analogue of the Kakeya conjecture would predict full Heisenberg Hausdorff dimension 4 for such sets.

We show that this prediction fails: their sharp lower bound is 4, and it remains sharp even among compact CC-geodesic Kakeya sets.

We also construct a CC-geodesic Kakeya set of full Heisenberg Hausdorff dimension 4 and zero Lebesgue measure.

Finally, for every $\kappa\in(0,2\pi]$, we construct a compact curvature-$\kappa$ Kakeya set of zero Lebesgue measure whose Euclidean and Heisenberg Hausdorff dimensions are both equal to 1.