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

Terence Tao

• Let H:=(100R10RR1) 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,d1−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 ℓ2 to be replaced by RD(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−cD), 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 RD, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius R≥2 in Γ:=(100Z10ZZ1) can be embedded into a bounded-dimensional space RD with the optimal distortion bound of O(log1/2R).