and $a_0 \in {\mathbb{Z}}$ we set $$\widetilde{{\mathbf{a}}} := (a_0, a_1, \dots, a_n).$$ We say that ${\mathbf{y}}\in {\mathbb{Q}}^{n}_S$ is $\Psi$-approximable (${\mathbf{y}}\in {\mathcal{W}}_{n}(S, \Psi)$) if there are infinitely many solutions ${\mathbf{a}}\in {\mathbb{Z}}^n$ to $$|a_0 + {\mathbf{a}}\cdot {\mathbf{y}}|_{S}^{l}  \leq \left\{
\begin{array}{rl} 
\Psi(\widetilde{{\mathbf{a}}}) & \text{ if } \infty \notin S\\ 
\\
\Psi({\mathbf{a}}) & \text{ if } \infty \in S.
\end{array} \right.$$

We fix Haar measure on ${\mathbb{Q}}_p$, normalized to give ${\mathbb{Z}}_p$ measure $1$ and denote the product measure on ${\mathbb{Q}}_S$ by $|~|_S$. Then, the following analogue of Khintchine’s theorem can be proved. Namely,

\[S-KG\] ${\mathcal{W}}_{n}(S, \psi)$ has zero or full measure depending on the convergence or divergence of the series $$\left\{ 
\begin{array}{rl} 
\sum_{k=1}^{\infty} k^{n}\psi(k) & \text{if } \infty \notin S \\ 
\\
 \sum_{k=1}^{\infty} k^{n-1}  \psi(k) & \text{if } \infty \in S.
\end{array} \right.$$

Indeed, the convergence case follows from the Borel-Cantelli lemma as usual and the divergence case can be proved using the methods in [@L].

Inhomogeneous approximation:
----------------------------

Given a multivariable approximating function $\Psi$ and a function $\theta : {\mathbb{R}}^n \to {\mathbb{R}}$, we set ${\mathcal{W}}^{\theta}_{n}(\Psi)$ to be the set of ${\mathbf{x}}\in {\mathbb{R}}^n$ for which there exist infinitely many ${\mathbf{a}}\in \mathbb{Z}^n\setminus \{\mathbf{0}\}$ such that $$\label{preKGinhom}
|a_0 +  {\mathbf{a}}\cdot {\mathbf{x}}+ \theta({\mathbf{x}})| < \Psi({\mathbf{a}})$$ for some 