Multislicing and effective equidistribution for random walks on some homogeneous spaces

Abstract

We consider a random walk on a homogeneous space $G/Λ$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x \in G/Λ$ which is not trapped in a finite $μ$-invariant set, the $n$-step distribution $μ^{*n}*δ_{x}$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist-Quint, whose equidistribution statements only hold in Cesàro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon which we call multislicing. The latter is a generalization of the discretized projection theorems à la Bourgain and we believe it presents independent interest.