Abstract
In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of 'weak non-planarity' of manifolds and more generally measures on the space M-m,M-n of m x n matrices over R is introduced and studied. This notion generalizes the one of non-planarity in R-n and is used to establish strong (Diophantine) extremality of manifolds and measures in M-m,M-n. Thus our results contribute to resolving a problem stated in [20, 9.1] regarding the strong extremality of manifolds in M-m,M-n. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results of the first named author and Velani [11] and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.