18 problems

Let $\mathbb{T}$ be a set of length-one $\delta$-tubes in $\mathbb{R}^n$, and let $U(\mathbb{T})$ denote their union. Assume that the tubes have $\delta$-separated directions. Kake…

Discretized sticky Kakeya conjecture. For every $\varepsilon>0$ there exist $\eta=\eta_{n,\mathrm{rest}}(\varepsilon)>0$ and $\delta_0=\delta_{0,n,\mathrm{rest}}(\varepsilon)>0$ su…

Curved Kakeya conjecture. If $\phi$ satisfies Bourgain's condition, then every $\phi$-Kakeya set has Hausdorff dimension $n$.

For $0<\delta<1$, let $K_\delta f(\omega)$ be the Kakeya maximal function, defined using averages over the $\delta$-neighborhoods of unit line segments in direction…

Let $L$ be a set of line segments in $\mathbb{R}^n$. For each line segment $\ell$, let $\widetilde\ell$ denote the line containing $\ell$. Keleti's line segment extension conjectur…

Let $n\geq 2$ and $\varepsilon>0$. For sufficiently small $\delta>0$, let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^n$, and let $\widetilde T$ denote the 2-fold dilate…

Let $\varepsilon>0$. A family $L$ of lines is directionally $\delta$-separated when its line directions are separated at scale $\delta$, and let $E\subset[0,1]^3$ be a union of…

Let $\mathcal L_{SL_2}$ be the set of lines in $\varmathbb{R}^3$ of the form $\ell_{(a,b,c,d)}=(a,b,0)+\operatorname{span}(c,d,1)$ with $ad-bc=1$, and let $\mathcal L_{SL_2}^*$ als…

Let $\mathcal{L}$ be the set of affine lines in $\mathbb{R}^n$, with packing dimension used for subsets of this line space. A compact set $K\subset\mathbb{R}^n$ is a sticky Kakeya…

$\Omega$-Kakeya maximal conjecture. Fix any probability measure $\mu$ on $\mathbb{S}^{n-1}$ satisfying, for some $d\in[0,n-1]$,

$\Omega$-Kakeya conjecture. For every Borel set $\Omega\subset\mathbb{S}^{n-1}$, if $X$ is an $\Omega$-Kakeya set, then

Let $N,n\in\mathbb{N}$, and let $S\subseteq (\mathbb{Z}/N\mathbb{Z})^n$ be a Kakeya set, meaning that $S$ contains a line in every direction in…

Let $K\subset\mathbb{R}^n$ be a Kakeya set, and let $G(n,k)$ denote the Grassmannian of $k$-dimensional subspaces of $\mathbb{R}^n$. For $\gamma\in G(n,k)$, write $\pi_\gamma$ for…

Let $\epsilon>0$ be any constant and let $q$ be a sufficiently large prime power. Let $L$ be a set of lines in $\mathbb{F}_q^3$ satisfying … and such that no plane contains more th…

Fix $n$, and let $I^n\subseteq\mathbb{R}^n$ be the unit cube. Given $\varepsilon,H>0$, a positive integer $N$, and a polynomial $Q$, subdivide $I^n$ into cubes of edge length…

Let $\pi_q$ be a projective plane of odd order $q$, and let $K$ be a Kakeya set in $\pi_q$, meaning that $K$ contains at least one point on each line of a fixed line pencil. Faber'…

Fixed-width Lipschitz Kakeya conjecture. For some $1<p<2$ and some finite $N$, for all $0<\delta<1$, all Lipschitz vector fields $v$, and…

Lipschitz Kakeya maximal-function conjecture. For some $1<p<2$ and some finite $N$, for all $0<\delta<1$ and all Lipschitz vector fields $v$, the operator…