Simultaneous inhomogeneous Diophantine approximation on manifolds

In 1998, Kleinbock & Margulis [KM98] established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent w0(x) = 1/n for almost every point x on a non-degenerate submanifold M of Rn. In this paper the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any ‘inhomogeneous’ vector θ ∈ Rn we prove that the simultaneous inhomogeneous Diophantine exponent w0(x,θ) = 1/n for almost every point x on M . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w0(x) = 1/n for almost all x ∈ M if and only if for any θ ∈ Rn the inhomogeneous exponent w0(x,θ) = 1/n for almost all x ∈ M. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in [BV]. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of [BV] while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.