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We give an integrability criterion on a real-valued non-increasing function 
$\unicode[STIX]{x1D713}$ guaranteeing that for almost all (or almost no) pairs 
$(A,\mathbf{b})$, where 
$A$ is a real 
$m\times n$ matrix and 
$\mathbf{b}\in \mathbb{R}^{m}$, the system is solvable in 
$$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$
$\mathbf{p}\in \mathbb{Z}^{m}$, 
$\mathbf{q}\in \mathbb{Z}^{n}$ for all sufficiently large 
$T$. The proof consists of a reduction to a shrinking target problem on the space of grids in 
$\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose 
$m=n=1$ case was recently solved, but whose general case remains open.
We study the distribution of the orbits of real numbers under the beta-transformation 
$T_{{\it\beta}}$ for any 
${\it\beta}>1$. More precisely, for any real number 
${\it\beta}>1$ and a positive function 
${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set: 
$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$
