Existence of a Kakeya set of zero Heisenberg Hausdorff measure in the first Heisenberg group (Liu's Problem 1.3)

In the first Heisenberg group $\mathbb{H}$ equipped with the Korányi metric, every Kakeya set (a set containing a left translate of every unit horizontal line segment) has Heisenberg Hausdorff dimension at least $3$, and this bound is sharp. Liu's paper posed, as its Problem 1.3, the measure-theoretic question at this critical dimension: does there exist a Kakeya set $E\subset\mathbb{H}$ with vanishing $3$-dimensional Heisenberg Hausdorff measure $\mathcal{H}^{3}_{\mathbb{H}}(E)=0$? A stronger related form asks whether there is a positive lower bound for $\mathcal{H}^{3}_{\mathbb{H}}$ among Kakeya sets. Despite subsequent activity on Heisenberg Kakeya maximal inequalities, dimension bounds, and Carnot–Carathéodory(CC)-geodesic variants (through 2026), this question remains open.

Background

A Kakeya set in $\mathbb{R}^{n}$ is a set containing a unit line segment in every direction; a classical construction shows such sets can have Lebesgue measure zero in every dimension, while the Kakeya conjecture (recently settled in $\mathbb{R}^{3}$) asserts they must nevertheless have full Hausdorff dimension. The first Heisenberg group $\mathbb{H}=\mathbb{H}^{1}$ is $\mathbb{R}^{3}$ endowed with the group law $(x,y,t)\cdot(x',y',t')=(x+x',\,y+y',\,t+t'+\tfrac{1}{2}(xy'-yx'))$ and the Korányi metric $d_{\mathbb{H}}(p,q)=\|q^{-1}p\|_{\mathbb{H}}$ with $\|(x,y,t)\|_{\mathbb{H}}=((x^{2}+y^{2})^{2}+16t^{2})^{1/4}$; the Hausdorff measure and dimension induced by $d_{\mathbb{H}}$ are denoted $\mathcal{H}^{s}_{\mathbb{H}}$ and $\dim^{\mathbb{H}}_{H}$ (balls of Korányi radius $r$ have Lebesgue volume comparable to $r^{4}$ — the homogeneous dimension is $4$ — although the space is topologically $3$-dimensional). Horizontal segments are unit segments issuing from the identity inside the plane $\{t=0\}$, i.e. the geodesics of the sub-Riemannian structure with zero curvature parameter. Liu (2021/2022) defined a Kakeya set in $\mathbb{H}^{n}$ as a set $E$ such that for every unit line segment $I\subset\mathbb{R}^{2n}\times\{0\}$ centered at the origin there exists $q\in\mathbb{H}^{n}$ with $qI\subset E$ (left translation implements the notion of 'parallel copy'). The key structural feature is that left-translating a horizontal segment by $q$ tilts it onto the graph $t=q_{3}+\tfrac{1}{2}(q_{1}y-q_{2}x)$: the translate remains horizontal only when $(q_{1},q_{2})$ is parallel to the segment's direction, and stays in the plane $\{t=0\}$ only when in addition $q_{3}=0$ — the case in which $\{t=0\}$ itself contains a translate of every unit horizontal segment, so that plane is a Kakeya set. For a generic translate this tilting is why the mechanism behind classical measure-zero Euclidean constructions is unavailable. Liu proved (Theorem 1.2) that every Kakeya set in $\mathbb{H}$ has $\dim^{\mathbb{H}}_{H}\geq 3$, sharply, with the horizontal plane itself as the extremal example; the proof combines a Marstrand-type restricted projection theorem in $\mathbb{R}^{3}$ with a co-area inequality via a duality encoding of horizontal segments. Later work recovered the same bound from a Heisenberg Kakeya maximal inequality (Fässler–Pinamonti–Wald) and computed the Euclidean dimension of unions of horizontal lines and so-called $SL(2)$ lines — lines $(a,b,0)+\mathrm{span}(c,d,1)\subset\mathbb{R}^{3}$ with $ad-bc=1$, which correspond to horizontal lines in $\mathbb{H}$ — (Fässler–Orponen), but all of these are dimension-level results. At the critical dimension $3$ the measure behavior is unsettled from both sides: no Kakeya set with $\mathcal{H}^{3}_{\mathbb{H}}(E)=0$ is known, and no positive lower bound on $\mathcal{H}^{3}_{\mathbb{H}}$ over Kakeya sets is known either.

Problem Statement

Does there exist a Kakeya set $E$ in the first Heisenberg group $\mathbb{H}$ (in the sense of Liu's Definition 1.1: for every unit horizontal line segment $I\subset\mathbb{R}^{2}\times\{0\}$ centered at the identity there is $q\in\mathbb{H}$ with $qI\subset E$) whose $3$-dimensional Heisenberg Hausdorff measure vanishes, $\mathcal{H}^{3}_{\mathbb{H}}(E)=0$, where $\mathcal{H}^{3}_{\mathbb{H}}$ is the Hausdorff measure induced by the Korányi metric? In the stronger form also raised in Liu's introduction: is there a positive lower bound for $\mathcal{H}^{3}_{\mathbb{H}}(E)$ among all Kakeya sets $E\subset\mathbb{H}$ (i.e. is $\inf\,\mathcal{H}^{3}_{\mathbb{H}}(E)$ over all Kakeya sets strictly positive), or can a Kakeya set be critical-dimension null? This is Problem 1.3 of Liu (arXiv:2106.11937 / Proc. Amer. Math. Soc. 2022), posed immediately after the theorem that every such Kakeya set has Heisenberg Hausdorff dimension at least $3$ with equality attained by the horizontal plane.

References

1. Jiayin Liu, On the dimension of Kakeya sets in the first Heisenberg group, Proceedings of the American Mathematical Society 150(8):3445–3455 (2022), DOI 10.1090/proc/15914, https://doi.org/10.1090/proc/15914 (preprint arXiv:2106.11937, https://arxiv.org/abs/2106.11937)
2. Katrin Fässler, Tuomas Orponen, A note on Kakeya sets of horizontal and SL(2) lines, Bulletin of the London Mathematical Society 55(5):2195–2204 (2023), DOI 10.1112/blms.12844, https://doi.org/10.1112/blms.12844 (preprint arXiv:2210.09955, https://arxiv.org/abs/2210.09955)
3. Katrin Fässler, Andrea Pinamonti, Pietro Wald, Kakeya maximal inequality in the Heisenberg group, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2023), article 289, DOI 10.2422/2036-2145.202301_015, https://doi.org/10.2422/2036-2145.202301_015 (preprint arXiv:2212.01845, 2022, https://arxiv.org/abs/2212.01845)
4. Marianna Chatzakou, A short note on the dimension of the CC-geodesic Kakeya sets in the first Heisenberg group, preprint (2026), arXiv:2607.26906, https://arxiv.org/abs/2607.26906
5. Alex Iosevich, Zhangze Li, Krystal Taylor, Positive Measure of Unions of Variable Surfaces, preprint (2026), arXiv:2605.27550, https://arxiv.org/abs/2605.27550