$a_0 \in {\mathbb{Z}}$. For $\psi$ as above, the set ${\mathcal{W}}^{\theta}_{n}(\psi)$ is often referred to as the (dual) set of “$(\psi, \theta)$-inhomogeneously approximable" vectors in ${\mathbb{R}}^n$. The following inhomogeneous version of Theorem \[KG\] is established in [@BaBeVe]. We denote by $C^n$ the set of $n$-times continuously differentiable functions.

\[KGinhom\] Let $\theta : \mathbb{R}^n \to \mathbb{R}$ be a $C^2$ function. Then $$|{\mathcal{W}}^{\theta}_{n}(\psi)| = \left\{
\begin{array}{rl}
0 & \text{if } \ \sum_{k=1}^{\infty}  k^{n-1}\psi(k) < \infty\\
\\
\text{ full } & \text{if } \ \sum_{k=1}^{\infty}  k^{n-1}\psi(k) = \infty.
\end{array} \right.$$

We remark that the choice of $\theta = \text{constant}$ is the setting of traditional inhomogeneous Diophantine approximation and in that case the above result was well known, see for example [@Cassels]. Similarly inhomogeneous Diophantine approximation can be considered in the $S$-arithmetic setting.

For a multivariable approximating function $\Psi$ and a function $\Theta: {\mathbb{Q}}^{n}_S \to {\mathbb{Q}}_S$, we say that a vector ${\mathbf{x}}\in  {\mathbb{Q}}_S^n $ is $(\Psi,\Theta)$-approximable if there exist infinitely many $({\mathbf{a}}, a_0)\in{\mathbb{Z}}^n\setminus\{0\}\times {\mathbb{Z}}$ such that $$|a_0 + {\mathbf{a}}\cdot {\mathbf{x}}+\Theta({\mathbf{x}})|_{S}^l\leq  \left\{
\begin{array}{rl} 
\Psi(\widetilde{{\mathbf{a}}}) & \text{ if } \infty \notin S\\ 
\\
\Psi({\mathbf{a}}) & \text{ if } \infty \in S.
\end{array} \right.$$

The convergence case of Khintchine’s theorem in this setting again follows from the Borel 