  A $(d,k)$-set is a subset of $\mathbb{R}^d$ containing a $k$-dimensional unit
ball of all possible orientations. Using an approach of D.~Oberlin we prove
various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest
is in restricted $(d,k)$-sets, where the set only contains unit balls with a
restricted set of possible orientations $\Gamma$. In this setting our estimates
depend on the Hausdorff dimension of $\Gamma$ and can sometimes be improved if
additional geometric properties of $\Gamma$ are assumed. We are led to consider
cones and prove that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension
$d-1$, which may be of interest in its own right.
