The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong extremality' of arbitrary finite collection of smooth nondegenerate submanifolds of ${ \bold R}^n$. The proofs are based on generalized quantitative nondivergence estimates for translates of measures on the space of lattices.
Categories: math.NT math.DS