more restrictive class of approximating functions. For an $n$-tuple $v = (v_1, \cdots, v_n)$ of positive numbers satisfying $v_1 + \cdots + v_n = n$, define the $v$-quasinorm $| ~ |_v$ on ${\mathbb{R}}^n$ by setting $$\|{\mathbf{x}}\|_v := \max |x_i|^{1/v_i}.$$ Following [@BaBeVe] we say that a multivariable approximating function $\Psi$ satisfies property $\mathbf{P}$ if $\Psi({\mathbf{a}}) = \psi(\|{\mathbf{a}}\|_v)$ for some approximating function $\psi$ and $v$ as above. As noted in loc. cit. when $v = (1, \dots, 1)$ we have that $\|{\mathbf{a}}\|_v = \|{\mathbf{a}}\|$ and any approximating function $\psi$ satisfies property $\mathbf{P}$, where $\psi$ is regarded as the function ${\mathbf{a}}\to \psi(\|{\mathbf{a}}\|)$. The proof of Theorem \[thm:divergence\] can be modified to deal with the case of functions satisfying property $\mathbf{P}$.

Structure of the paper {#structure-of-the-paper .unnumbered}
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In the next section, we recall the transference principle of Beresnevich and Velani. The subsequent section studies $(C, \alpha)$-good functions in the $S$-adic setting. We then prove Theorem \[thm:main\] and then Theorem \[thm:divergence\]. We conclude with some open questions.

Inhomogeneous transference principle
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In this section we state the inhomogeneous transference principle of Beresnevich and Velani from [@BeVe Section 5] which will allow us to convert our inhomogeneous problem to the homogeneous one. Let $(\Omega, d)$ 