Simultaneous inhomogeneous Diophantine approximation on manifolds

In 1998, Kleinbock and Margulis proved Sprindzuk's conjecture pertaining to metrical Diophantine approximation (and indeed the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_0(\mathbf x)=1/n$ for almost every point $\mathbf x$ on a nondegenerate submanifold $\mathcal M$ of $\mathbb R^n$. In this paper, the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any “inhomogeneous” vector $\boldsymbol\theta\in\mathbb R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_0(\mathbf x,\boldsymbol\theta)$ is $1/n$ for almost every point $\mathbf x$ on $\mathcal M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\mathbf x)$ is $1/n$ for almost all $\mathbf x\in\mathcal M$ if and only if, for any $\boldsymbol\theta\in\mathbb R^n$, the inhomogeneous exponent $w_0(\mathbf x,\boldsymbol\theta)=1/n$ for almost all $\mathbf x\in\mathcal M$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.