Decorrelation estimates for translated measures under diagonal flows

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.