Fourier analytic variants of the Furstenberg and Kakeya problems

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K\subseteq {\mathbb{R}}^2$ an $(s,t)$-Kakeya set if there exists a set of directions $E\subseteq S^1$ with Hausdorff dimension at least $t$ such that, for each $e\in E$, the set $K$ contains a subset of a unit line segment in direction $e$ whose Fourier dimension, viewed as a subset of $\mathbb{R}$, is at least $s$. For $\mathrm{\Delta }(s,t)$ defined to be the infimum of the Fourier dimension among all $(s,t)$-Kakeya sets in ${\mathbb{R}}^2$, we prove that $$\frac{2st}{s+2t}\le \mathrm{\Delta }(s,t)\le \min \{s,2t\}\mathrm{.}$$ These bounds, though distinct, are asymptotically equivalent as either $s$ or $t$ tends to zero. We also obtain upper and lower bounds in the Furstenberg set version of the problem and in the case where the Hausdorff dimension of the collection of lines is replaced by the Fourier dimension.