15 problems

Fix $n\geq 2$ and $0<\delta\ll 1$. Let $\Omega$ be a maximal $\delta$-separated subset of directions on $S^{n-1}$, and for each $\omega\in\Omega$ let $T_\omega$ be a…

Kakeya conjecture. For every $\varepsilon>0$,

Let $R$ range over finite sets of rational slopes, let $s$ be an output slope, and let ${\operatorname{SD}}(R;s)$ denote the associated sum-difference exponent. Arithmetic Kakeya c…

Let $0<\delta\leq\nu<1$. Three families $\mathbb{T}_{1},\mathbb{T}_{2},\mathbb{T}_{3}$ of $\delta$-tubes are $\nu$-disjoint when their orientations lie in spherical patches of diam…

Let $0<\delta<1$. A $\delta$-tube is a tube of length $1$ and cross-sectional diameter $\delta$, and a family $\mathbb{T}$ of $\delta$-tubes is $\delta$-separated when distinct tub…

Let $0<\varepsilon<1$, and let $\mathcal{K}^{\ast}(\otimes_1L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon)$ be the statement that for every $0<\delta<1$ and every family…

Let $E$ be an arbitrary subset of $\mathbb{R}^n$, and let $D$ be the set of directions in which $E$ contains a line segment. General Kakeya conjecture. If $D$ is non-empty, then ……

Let $\mathcal{S}$ be any collection of line segments in Euclidean space, and let $\mathcal{L}$ be the collection of the corresponding full lines. Line-segment extension conjecture.…

X-ray endpoint conjecture. For every $\varepsilon>0$, there is a constant $C_\varepsilon<\infty$ such that

Let $k$ be a positive integer. For $N$ different integer common differences, let $F'_k(N)$ be the cardinality of the smallest set $A\subset\mathbf{Z}$ containing a $k$-term arithme…

Let $k$ be a positive integer and let $p$ be prime. Write $f_{k,n}(p)$ for the size of the smallest set containing, for every nonzero $d\in\mathbf{F}_p^n$, a $k$-term progression w…

Katz–Tao projection conjecture. For every $\varepsilon>0$, there are $r_1,\dots,r_k\in\mathbf{Q}\cup\\{\infty\\}$, none equal to $-1$, such that

Entropy arithmetic Kakeya conjecture. For every $\varepsilon>0$, there are $r_1,\dots,r_k\in\mathbf{Q}$, none equal to $-1$, such that

Let $k, N$ be positive integers. Write $F_k(N)$ for the size of the smallest set of integers containing, for each $d \in \\{1,\dots, N\\}$, a $k$-term arithmetic progression with c…

Let $d=2n+1$, and let … be the $d$-dimensional hyperbolic paraboloid. For a function $f:\mathbb{F}^n\times\mathbb{F}^n\to\mathbb{C}$, define … where…