On inhomogeneous Diophantine approximation and Hausdorff dimension

Let $\mathrm{\Gamma }=ZA+Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let $\omega (A)$ denote the exponent of uniform simultaneous rational approximation to the point $A$. We show that for any real number $v\ge \omega (A)$, the Hausdorff dimension of the set $B_v$ of points in $R^n$ which are $v$-approximable with respect to $\mathrm{\Gamma }$, is equal to $1\mathrm{/}v$.