  Let $\Gamma = Z A +Z^n$ 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 $\Gamma$, is equal to $1/v$.
