In 2004, T. Tao introduced a gauge transform which was widely used afterwards by several authors in the study of wellposedness at low regularity. In this talk, I will explain how this gauge transform arises as a high frequency approximation of the nonlinear Fourier transform recently associated to the integrable structure of this equation on the torus. As an application, I will describe the dynamics of the Benjamin-Ono equation up to smoother remainders.

This a jointwork with T. Kappeler and P. Topalov.