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

On the Fourier dimension of $(d,k)$-sets and Kakeya sets with restricted directions