2024/09/05 by Timothée Bénard, Bénard, Timothée, Weikun He +1 · 1 citation
Computer Science · Mathematics · #51B99 #60G50 #Classical Analysis and ODEs (math.CA) #Data Management and Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 37A99 #Secondary 22E99 #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2409.03300
openalex publication_date 2024/09/05 · openalex created_date 2024/10/23 · openalex updated_date 2026/07/28
We consider a random walk on a homogeneous space G/Λ where G is SO(2,1) or 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 ∈ G/Λ which is not trapped in a finite μ-invariant set, the n-step distribution μ*n*δx 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.