24 problems

Polynomial Wolff axioms conjecture. The following assertions should hold:

For $j=1,\ldots,n$, let $\mathcal{T}_j$ be a family of $\delta$-tubes in $\mathbb{R}^n$. Call the families transversal when, for each $j$, every tube in $\mathcal{T}_j$ points in a…

Guth–Zahl conjecture. Let $p\geqslant\frac{n}{n-1}$. For every $\varepsilon>0$, there are a complexity $D=D_{\varepsilon,n}\in\mathbb{N}$ and a constant $C_{\varepsilon,n}>0$ such…

$(n,k)$ maximal-function conjecture. For all $\epsilon>0$ and $0<\delta\ll1$,

$(n,k)$-set dimension conjecture. If $E\subset\mathbb{R}^n$ is an $(n,k)$ set, then

For an odd prime power $q$, a Besicovitch set in ${\mathbb{F}_q}^2$ is a subset containing a line in every direction. Finite-plane Besicovitch set conjecture. The smallest Besicovi…

Multiplicity-line conjecture. If $B$ is small, then there is some $j\in\mathbb{F}_q\cup\{\infty\}$ such that every point $P\in B$ with $m_P\geq3$ lies on the line $\ell(j,b_j)$. Th…

Let $q$ be odd, and write $\ell(m,b)$ and $\ell(\infty,a)$ for the lines $y=mx+b$ and $x=a$ in $\mathbb{F}_q^2$. A Besicovitch set is a set $B\subset\mathbb{F}_q^2$ containing one…

Let $n\geq 2$, let $p=2n/(n+1)$, and let the oscillatory sum and square function be the quantities denoted by and in the source. Endpoint oscillatory sum conjecture. One should be…

Let $Z$ be a real vector space. For a slope $r\in\mathbb{R}$, define $\pi_r:Z\times Z\to Z$ by $\pi_r(a,b)=a+rb$, and define $\pi_\infty(a,b)=b$. For a finite collection $R$ of pro…

Let $E \subseteq \mathbb{R}^n$, with $n>1$, be a Besicovitch set, meaning that $E$ contains a unit line segment in every direction. Its upper Minkowski dimension is the upper Minko…

Let the bilinear tube estimate in Theorem $1$ be understood with its stated $L^p$ exponent, and let the sharp exponent mean the smallest exponent for which that estimate holds. Sha…

Discretized Kakeya conjecture. The following two assertions should hold:

Let $n=3$, let $\Sigma$ be a strictly convex hypersurface, and let $\mathop{\mathrm{MKNE}}$ denote the maximal Kakeya--Nikodym estimate referred to in the source as equation (MKNE)…

An $SL_2$ Besicovitch set in $\mathbb{R}^3$ is a Besicovitch set associated with the $SL_2$ example arising in the Kakeya problem. Wang–Zahl's conjecture. Every $SL_2$ Besicovitch…

The family of fractal Fourier restriction estimates. For every $\epsilon>0$ there is a constant $C_\epsilon$ such that

Fix a positive integer $n$. Let $\mathcal{f}^M_n(k)$ be the infimum of the Minkowski dimension of a set $B_k\subset\mathbb{R}^n$ containing a $k$-term arithmetic progression with e…

Fix a positive integer $n$. Let $F'_{n,k}(N)$ be the minimum cardinality of a set containing a $k$-term arithmetic progression in $\mathbb{R}^n$ with $N^n$ distinct common differen…

Maximal tubular incidence conjecture. Let ${l_i}_{i\in\{1,\dots,L\}}$ be $L$ unit line segments in $\mathbb{R}^3$ whose directions are $\delta$-separated. Then there is a constant…

Low-degree variety concentration conjecture. The third condition holds for every collection of tubes pointing in $\delta$-separated directions.

Direction-separated polynomial Wolff conjecture. Every set of tubes pointing in $\delta$-separated directions satisfies the polynomial Wolff axioms; more precisely, it satisfies th…

Wolff-axiom volume conjecture. In three dimensions, the union of any set of tubes satisfying the Wolff axioms has volume close to $1$.

Kakeya conjecture for the polynomial Wolff axioms. For every dimension $n$, there is a complexity $E$ such that, whenever $\mathbb{T}$ is a set of $\delta$-tubes in $\mathbb{R}^n$…

Let $L(t_1,t_2),M(t_1,t_2)\in\mathbb{F}_{q_0}[t_1,t_2]$ be arbitrary polynomials. Consider the map … For each extension $\mathbb{F}_q/\mathbb{F}_{q_0}$, let $E_{\mathbb{F}_q}$ be t…