by Beresnevich and Velani in the form used in [@BaBeVe]. The structure of the proof is the same as in [@BaBeVe]. We also take the opportunity to clarify some properties of $(C, \alpha)$-good functions in the $S$-adic setting which may be of independent interest.

3.  The proof of Theorem \[thm:divergence\], follows the ubiquity framework used in [@BaBeVe] but needs new ideas to implement in the $p$-adic setting. At present, we are unable to prove the more general $S$-adic divergence statement. We note that the $S$-adic case remains open even in the homogeneous setting.

4.  We now undertake a brief discussion of the assumptions (I1) - (I5). The conditions (I1)-(I4) are assumed in [@MoS1] and, as explained in loc. cit., are assumed for convenience. Namely, as mentioned in [@MoS1], the statement for any non-degenerate analytic manifold over ${\mathbb{Q}}_S$ follows from Theorem \[thm:main\]. In [@BaBeVe], the inhomogeneous parameter $\Theta$ is allowed to be $C^2$ when restricted to the nondegenerate manifold. However, we need to assume it to be analytic.

5.  Theorem \[thm:divergence\] is slightly more general than Theorem 1.2 of [@MoS2] in the homogeneous setting. In [@MoS2], the approximating function is taken to be of the form $$\Psi({\mathbf{a}})=\frac{1}{\|{\mathbf{a}}\|^{n}}\psi(\|{\mathbf{a}}\|), {\mathbf{a}}\in{\mathbb{Z}}^{n+1}$$ which is a 