Decorrelation estimates for translated measures under diagonal flows

Journal article, 2025

A profound link between homogeneous dynamics and Diophantine approximation is based on an observation that Diophantine properties of a real matrix B are encoded by the corresponding lattice ΛB translated by a multi-parameter semigroup a(t). We establish quantitative decorrelation estimates for measures supported on leaves a(t)ΛB with the error terms depending only on the minimum of the pairwise distances between the parameters. The proof involves a careful analysis of the translated measures in the products of the spaces of unimodular lattices and establishes quantitative equidistributions to measures supported on various intermediate homogeneous subspaces.