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

Abstract

Let $H:=\left(\begin{array}{ccc}1& \mathbb{R}& \mathbb{R}\\ 0& 1& \mathbb{R}\\ 0& 0& 1\end{array}\right)$ denote the Heisenberg group with the usual Carnot–Carathéodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0<\epsilon \le 1/2$, the snowflaked metric space $(H,d^{1-\epsilon })$ embeds into an infinite-dimensional Hilbert space with distortion $O({\epsilon }^{-1/2})$. 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 ${\mathbb{R}}^{D(\epsilon )}$ for some dimension $D(\epsilon )$ dependent on $\epsilon$. Naor and Neiman showed that $D$ could be taken independent of $\epsilon$, at the cost of worsening the bound on the distortion to $O({\epsilon }^{-1-c_D})$, where $c_D\to 0$ as $D\to \infty$. In this paper we show that one can in fact retain the optimal distortion bound $O({\epsilon }^{-1/2})$ and still embed into a bounded-dimensional space ${\mathbb{R}}^D$, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius $R\ge 2$ in $\Gamma :=\left(\begin{array}{ccc}1& \mathbb{Z}& \mathbb{Z}\\ 0& 1& \mathbb{Z}\\ 0& 0& 1\end{array}\right)$ can be embedded into a bounded-dimensional space ${\mathbb{R}}^D$ 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.