quotient $\Phi_\beta$ and $x_\nu,y_1,y_2,y_3 \in U_\nu$.

We can now state the first main Theorem of the present paper.

\[thm:main\] Let $S$ be as in (I0) and ${\mathbf{U}}$ as in (I1). Suppose ${\mathbf{f}}$ satisfies (I2) and (I3), that $\Psi$ satisfies (I4) and $\Theta$ satisfies (I5). Then $${\mathcal{W}}_{\Psi,\Theta}^{{\mathbf{f}}} := \{ {\mathbf{x}}\in{\mathbf{U}}| \ {\mathbf{f}}({\mathbf{x}})    \text{  is   }     (\Psi,\Theta)-\text{  approximable}\}$$ has measure zero if $\sum_{{\mathbf{a}}\in {\mathbb{Z}}^n\setminus\{0\}} \Psi({\mathbf{a}}) <\infty$.

The divergence case of our Theorem is proved in the more general setting of Hausdorff measures. However, we need to impose some restrictions: we only consider the case when $S = \{p\}$ consists of a single prime, the inhomogeneous function is assumed to be analytic, and the approximating function is not as general as in Theorem \[thm:main\]. We will denote by $\mathcal{H}^{s}(X) $ the $s$-dimensional Hausdorff measure of a subset $X$ of ${\mathbb{Q}}^{n}_{S}$ and $\dim X$ the Hausdorff dimension, where $s > 0$ is a real number.

\[thm:divergence\] Let $S$ be as in (I0) and ${\mathbf{U}}$ as in (I1). Suppose ${\mathbf{f}}:{\mathbf{U}}\subset{\mathbb{Q}}_p^m\to {\mathbb{Q}}_p^n$ satisfies (I2) and (I3). Let $$\label{def:newpsi}
        \Psi({\mathbf{a}})= \psi(\|{\mathbf{a}}\|), {\mathbf{a}}\in{\mathbb{Z}}^{n+1}$$ be an approximating function and assume that $s > m-1$. Let $\Theta:{\mathbf{U}}\to {\mathbb{Q}}_p$ be an analytic map satisfying (I5). Then $$\mathcal{H}^s(\mathcal{W}^{\mathbf{f}}_{(\Psi,\Theta)}\cap{\mathbf{U}})=\mathcal{H}^s({\mathbf{U}}) \text{    if   } \sum_{{\mathbf{a}}\in 