Caltech/UCLA Joint Analysis Seminar

From the work of Pansu and Semmes it is known that the Heisenberg group (with the Carnot-Caratheodory metric) cannot embed into Euclidean space (or even Hilbert space) in a bilipschitz fashion. However if one "snowflakes" the metric then this becomes possible thanks to work of Assouad. There is a lower bound on the distortion in doing so due to Austin, Naor, and Tessera; we show that this bound can be attained while embedding into a bounded dimensional Euclidean space, answering a question of Naor and Neiman in the negative. Our argument uses an iteration inspired by the Nash-Moser iteration scheme.