Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion
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 $0<\varepsilon<1/2$ the metric space $(G,d_G^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^{D_G}$ with distortion $O_G(\varepsilon^{-1/2})$. This is done by building on the approach of T. Tao (2021), who established the above assertion when …