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

### Terence Tao

University of California Los Angeles, USA

## Abstract

Let $H:=(100 R10 RR1 )$ 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 $0<ε≤1/2$, the snowflaked metric space $(H,d_{1−ε})$ embeds into an infinite-dimensional Hilbert space with distortion $O(ε_{−1/2})$. This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group $H$. Assouad's argument allows $ℓ_{2}$ to be replaced by $R_{D(ε)}$ for some dimension $D(ε)$ dependent on $ε$. Naor and Neiman showed that $D$ could be taken independent of $ε$, at the cost of worsening the bound on the distortion to $O(ε_{−1−c_{D}})$, where $c_{D}→0$ as $D→∞$. In this paper we show that one can in fact retain the optimal distortion bound $O(ε_{−1/2})$ and still embed into a bounded-dimensional space $R_{D}$, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius $R≥2$ in $Γ:=(100 Z10 ZZ1 )$ can be embedded into a bounded-dimensional space $R_{D}$ with the optimal distortion bound of $O(g_{1/2}R)$.

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.

## Cite this article

Terence Tao, Embedding the Heisenberg group into a bounded-dimensional Euclidean space with optimal distortion. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 1–44

DOI 10.4171/RMI/1200