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

A (d, k)-set is a subset of R 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 Γ. In this setting our estimates depend on the Hausdorff dimension of Γ and can sometimes be improved if additional geometric properties of Γ are assumed. Mathematics Subject Classification 2010: primary: 42B10, 28A80, 28A78.

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