42 problems

A Kakeya set in $\mathbf{R}^d$ is a subset containing a unit line segment in every direction. The Hausdorff dimension of a set is its Hausdorff dimension in the ambient Euclidean s…

Wolff's finite-field Kakeya conjecture. There is a constant $c_n>0$, depending only on $n$, such that every Kakeya set $K\subset\mathbb{F}_q^n$ satisfies

Finite-field Kakeya conjecture. Such a Kakeya set should have cardinality comparable to that of the ambient space, namely $|K|\approx|\mathbb{F}_q^n|=q^n$ up to a multiplicative co…

A $(d,k)$ set is a measurable subset of $\mathbb{R}^d$ containing a translate of every $k$-dimensional plane. For $k\geq 2$, a $(d,k)$ Besicovitch set is a $(d,k)$ set of measure z…

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…

A Kakeya set in $\mathbb{R}^3$ is a compact set $K\subset\mathbb{R}^3$ that contains a unit line segment pointing in every direction. The three-dimensional Kakeya conjecture. Every…

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…

Local $k$-plane transform conjecture. If

Mixed-norm x-ray transform conjecture. Together with the condition $r<\infty$, these necessary conditions are also sufficient for the estimate to hold. This conjecture asks for the…

A Besicovitch set is a measure-zero $(d,1)$ set, namely a subset of $\mathbb{R}^d$ containing a translate of every line. Besicovitch-set dimension conjecture. Every Besicovitch set…

A Besicovitch set is a measurable subset of $\mathbb{R}^d$ containing a translate of every line in $\mathbb{R}^d$. Besicovitch set dimension conjecture. Every Besicovitch set has H…

Let $1<l<n$, let $E\subset\mathbb R^n$ be a Borel set, and let $G(n,l)$ denote the Grassmannian of $l$-dimensional linear subspaces of $\mathbb R^n$. For $V\in G(n,l)$, write…

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…

Sticky curved Kakeya conjecture. If $\phi$ satisfies Bourgain's condition, then sticky $\phi$-Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$.

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

Let $\phi$ be a H"ormander phase function satisfying Bourgain's condition at every point. Let $p$ be the exponent in the restriction estimate … should hold for all … The conjecture…

Let $E\subset\mathbb R^d$ be a non-sticky Kakeya set, and let $m_d$ denote $d$-dimensional Lebesgue measure. Positive-measure conjecture for non-sticky Kakeya sets. Every non-stick…

Let $T_N$ be an oscillatory integral operator with a phase function satisfying Hörmander's non-degeneracy condition, acting on functions supported in $\mathbb{B}_1^{d-1}(0)$. Hörma…

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…

A Kakeya set in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. A Kakeya set is sticky in the sense formalized by Wang and Zahl when it satisfies…

The $\delta^n$ lower-bound conjecture. The normalized Haar measure of $S$ is at least

The real Kakeya dimension conjecture. Such a set $K$ has Minkowski and Hausdorff dimension $n$.

The $(n,k)$-Besicovitch set nonexistence conjecture. $(n,k)$-Besicovitch sets do not exist for $k\ge 2$.

Let $K\subset\mathbb{R}^n$ be a Kakeya set. Say that $K$ is $\varepsilon$-close to being sticky if there is a set of lines $L$ with packing dimension at most $n-1+\varepsilon$, con…