Decorrelation estimates for translated measures under diagonal flows

Abstract

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 ${\Lambda }_B$ translated by a multi-parameter semigroup $a(t)$. We establish quantitative decorrelation estimates for measures supported on leaves $a(t){\Lambda }_B$ 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.