EMBEDDING SNOWFLAKES OF CARNOT GROUPS INTO BOUNDED DIMENSIONAL EUCLIDEAN SPACES WITH OPTIMAL DISTORTION

Abstract

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any the metric space $(G,d_G^{1-𝜀})$ admits a bi-Lipschitz embedding into ${ℝ}^{D_G}$ with distortion $O_G({𝜀}^{-1∕2})$. We do this by building on the approach of T. Tao (Rev. Mat.Iberoam. 37:1 (2021), 1–44), 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 Lovász local lemma and the concentration of measure phenomenon on the sphere in place of Tao’s use of a quantitative homotopy argument.