Polynomially effective equidistribution for unipotent orbits in products of ${\mathrm{SL}}_2$ factors

Abstract

We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf{K}$-forms of ${\mathrm{SL}}_2^{\mathsf{n}}$ where $\mathbf{K}$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The key new phenomenon is the existence of many intermediate groups between the ${\mathrm{SL}}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions.

10 pages, research announcement