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

Abstract

Let H := \Big(\begin{smallmatrix} 1 & \mathbb{R} & \mathbb{R} \\ 0 & 1 & \mathbb{R} \\ 0 & 0 & 1 \end{smallmatrix}\Big) 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 < \varepsilon \leq 1/2 , 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 \mathbb{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 \mathbb{R}^D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius R \geq 2 in \Gamma := \Big(\begin{smallmatrix} 1 & \mathbb{Z} & \mathbb{Z} \\ 0 & 1 & \mathbb{Z} \\ 0 & 0 & 1 \end{smallmatrix}\Big) 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.