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

Abstract

 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 cD → 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 radiusThe 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.