We show that the Heisenberg group $\mathbb{ H}^n$ contains a measure zero set $N$ such that every Lipschitz function $f\colon \mathbb{ H}^n \to \mathbb{ R}$ is Pansu differentiable at a point of $N$. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of $N$ and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.
Categories: math.FA