On inhomogeneous Diophantine approximation and Hausdorff dimension

Let $\Gamma=\mathbf ZA+\mathbf Z^n\subset\mathbf R^n$ be a dense subgroup of rank $n+1$ and let $\hat\omega(A)$ denote the exponent of uniform simultaneous rational approximation to the generating point $A$. For any real number $v\ge\hat\omega(A)$, the Hausdorff dimension of the set $\mathcal B_v$ of points in $\mathbf R^n$ that are $v$-approximable with respect to $\Gamma$ is shown to be equal to $1/v$.