  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.
