Let $H := \begin{pmatrix} 1 & {\mathbf R} & {\mathbf R} \\ 0 & 1 &{\mathbf R} \\ 0 & 0 & 1 \end{pmatrix}$ denote the Heisenberg group with the usual Carnot-Carath\'eodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipchitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \varepsilon < 1$, the snowflaked metric space $(H,d^{1-\varepsilon})$ embeds into an infinite-dimensional Hilbert space with distortion $O( \varepsilon^{-1/2} )$. This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group $H$. Assouad's argument allows $\ell^2$ to be replaced by ${\mathbf R}^{D(\varepsilon)}$ for some dimension $D(\varepsilon)$ dependent on $\varepsilon$. Naor and Neiman showed that $D$ could be taken independent of $\varepsilon$, at the cost of worsening the bound on the distortion to $O( \varepsilon^{-1+c_D} )$, where $c_D 查看全文>>