Non-planarity and metric Diophantine approximation for systems of linear forms

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 spaceMm,n ofm × nmatrices over ℝ is introduced and studied. This notion generalizes the one of non-planarity in ℝⁿand is used to establish strong (Diophantine) extremality of manifolds and measures inMm,n . Thus our results contribute to resolving a problem stated in [20, §9.1] regarding the strong extremality of manifolds inMm,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.