field and $U$ is an open subset of $F^d$, and say that $f$ is nondegenerate at $x_0 \in U$ if the space $F^n$ is spanned by partial derivatives of $f$ at $x_0$ up to some finite order. Loosely speaking, a nondegenerate manifold is one in which is locally not contained in an affine subspace. Subsequent to the work of Kleinbock and Margulis, there were rapid advances in the theory of dual approximation on manifolds. In [@BKM] (and independently in [@Ber1]) the convergence case of the Khintchine-Groshev theorem for nondegenerate manifolds was proved and in [@BBKM], the complementary divergence case was established.

As for the $p$-adic theory, Sprindžuk [@Sp] himself established the $p$-adic and function field (i.e. positive characteristic) versions of Mahler’s conjectures. Subsequently, there were several partial results (cf. [@Kov; @BK]) culminating in the work of Kleinbock and Tomanov [@KT] where the $S$-adic case of the Baker-Sprindžuk conjectures were settled in full generality. In [@G], the second named author established the function field analogue. The convergence case of Khintchine’s theorem for nondegenerate manifolds in the $S$-adic setting was established by Mohammadi and Golsefidy [@MoS1] and the divergence case for ${\mathbb{Q}}_p$ in [@MoS2].

In the case of inhomogeneous Diophantine approximation on 