The linear Kakeya conjecture

For $0<\delta\ll1$, let $\mathbb{T}$ be a collection of rectangular $\delta$-tubes in $\mathbb{R}^d$, each with $d-1$ sides of length $\delta$ and one side of length $1$, whose orientations form a $\delta$-separated subset of $\mathbb{S}^{d-1}$. Let $\chi_T$ denote the indicator of $T$ and let $p'$ be the Hölder conjugate of $p$. Linear Kakeya conjecture. For every $\varepsilon>0$, if $\frac{1}{q}\leq\frac{d-1}{d}$ and $\frac{d-1}{p}+\frac{1}{q}\leq d-1$, then there is a constant $C_\varepsilon$, independent of $\delta$ and $\mathbb{T}$, such that

$$\left\|\sum_{T\in\mathbb{T}}\chi_T\right\|_{L^q(\mathbb{R}^d)}\leq C_\varepsilon\delta^{\frac{d}{q}-\frac{d-1}{p'}-\varepsilon}(\#\mathbb{T})^{\frac{1}{p}}.$$

This conjecture implies the Kakeya set conjecture, that every Borel set containing a unit line segment in every direction has full Hausdorff dimension.