  The inhomogeneous Groshev type theory for dual Diophantine approximation on
manifolds is developed. In particular, the notion of nice manifolds is
introduced and the divergence part of the theory is established for all such
manifolds. Our results naturally incorporate and generalize the homogeneous
measure and dimension theorems for non-degenerate manifolds established to
date. The generality of the inhomogeneous aspect considered within enables us
to make a new contribution even to the classical theory in R^n. Furthermore,
the multivariable aspect considered within has natural applications beyond the
standard inhomogeneous theory such as to Diophantine problems related to
approximation by algebraic integers.
