$S$-arithmetic Inhomogeneous Diophantine approximation on manifolds

We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For $S$ consisting of more than one prime, finite or infinite, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprind\v{z}uk conjectures using nondivergence estimates for flows on homogeneous spaces in conjunction with the transference principle of Beresnevich and Velani.