JournalsrmiVol. 37, No. 1pp. 1–44

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

  • Terence Tao

    University of California Los Angeles, USA
Embedding the Heisenberg group into a bounded-dimensional Euclidean space with optimal distortion cover

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Abstract

Let H:=(1RR01R001)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 dd. It is known (since the work of Pansu and Semmes) that the metric space (H,d)(H,d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0<ε1/20 < \varepsilon \leq 1/2, the snowflaked metric space (H,d1ε)(H,d^{1-\varepsilon}) embeds into an infinite-dimensional Hilbert space with distortion O(ε1/2)O( \varepsilon^{-1/2} ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group HH. Assouad's argument allows 2\ell^2 to be replaced by RD(ε)\mathbb{R}^{D(\varepsilon)} for some dimension D(ε)D(\varepsilon) dependent on ε\varepsilon. Naor and Neiman showed that DD could be taken independent of ε\varepsilon, at the cost of worsening the bound on the distortion to O(ε1cD)O( \varepsilon^{-1-c_D} ), where cD0c_D \to 0 as DD \to \infty. In this paper we show that one can in fact retain the optimal distortion bound O(ε1/2)O( \varepsilon^{-1/2} ) and still embed into a bounded-dimensional space RD\mathbb{R}^D, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius R2R \geq 2 in Γ:=(1ZZ01Z001)\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 RD\mathbb{R}^D with the optimal distortion bound of O(log1/2R)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.

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