Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any $0<\epsilon <1\mathrm{/}2$ the metric space $(G,d_G^{1-\epsilon })$ admits a bi-Lipschitz embedding into ${\mathbb{R}}^{D_G}$ with distortion $O_G({\epsilon }^{-1\mathrm{/}2})$. This is done by building on the approach of T. Tao (2021), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash–Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lov'{a}sz local lemma and the concentration of measure phenomenon on the sphere in place of Tao’s use of a quantitative homotopy argument.