Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion

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 $c_D \to 0$ as $D \to \infty$. In this paper we show that one can in fact retain the optimal distortion bound $O( \varepsilon^{-1/2} )$ and still embed into a bounded dimensional space ${\mathbf R}^D$, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius $R \geq 2$ in $\Gamma := \begin{pmatrix} 1 & {\mathbf Z} & {\mathbf Z} \\ 0 & 1 & {\mathbf Z} \\ 0 & 0 & 1 \end{pmatrix}$ can be embedded into a bounded-dimensional space ${\mathbf R}^D$ with the optimal distortion bound of $O(\log^{1/2} R)$. The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain ``loss of derivatives'' problem in the iteration.