such that $$\label{preKG}
|a_0 +  {\mathbf{a}}\cdot {\mathbf{x}}| < \Psi({\mathbf{a}})$$ for some $a_0 \in {\mathbb{Z}}$. When $\Psi({\mathbf{a}}) = \psi(\|{\mathbf{a}}\|)$ for a non-increasing function $\psi$, we write ${\mathcal{W}}_{n}(\psi)$ for ${\mathcal{W}}_{n}(\Psi)$. Khintchine’s Theorem ([@Khintchine], [@Groshev]) gives a characterization of the measure of ${\mathcal{W}}_{n}(\psi)$ in terms of $\psi$:

\[KG\] $$|{\mathcal{W}}_{n}(\psi)| = \left\{ 
\begin{array}{rl} 
0 & \text{if } \sum_{k=1}^{\infty} k^{n-1} \psi(k) < \infty\\ 
\\
\text{ full } & \text{if } \sum_{k=1}^{\infty}k^{n-1}  \psi(k) = \infty.
\end{array} \right.$$

Here, $\|~\|$ denotes the supremum norm of a vector and $|~|$ denotes the absolute value of a real number as well as the Lebesgue measure of a measurable subset of ${\mathbb{R}}^n$; the context will make the use clear. The kind of approximation considered above is called “dual" approximation in the literature as opposed to the setting of simultaneous Diophantine approximation. In this paper, we will only consider dual approximation. Given an approximation function, one can consider the corresponding $S$-arithmetic question as follows, we follow the notation of Kleinbock and Tomanov [@KT]. Given a finite set of primes $S$ of cardinality $l$ we set ${\mathbb{Q}}_S := \prod_{\nu \in S}{\mathbb{Q}}_\nu$ and denote by $|~|_S$ the $S$-adic absolute value, $|{\mathbf{x}}| = \max_{v \in S }|x^{(v)}|_v$. For ${\mathbf{a}}= (a_1, \dots, a_n) \in {\mathbb{Z}}^n$ 