36 problems

Finite-field Erdős distance conjecture. If

Let $S$ be a $k$-distance set in an $n$-dimensional normed space, meaning that the set of distances between distinct points of $S$ contains at most $k$ numbers. $k$-distance set co…

Let $Q_0,Q_1,Q_2$ be three cubes in $\mathbb{B}(0,C)$ of radius $\approx1$ satisfying … for all $x_0\in Q_0$, $x_1\in Q_1$, and $x_2\in Q_2$. For each $j=0,1,2$, let $E_j$ be a…

For a compact set $K\subset\mathbb{R}^2$, let $\operatorname{dist}(K)=\{|x-y|:x,y\in K\}$ be its distance set, and let ${\text{\roman dim}}$ denote Hausdorff dimension. Distance-se…

Affine Fourier-spectrum threshold conjecture. If, for some $\theta\in[0,1]$,

The realization conjecture. There exists a finite ultrametric space $(X,d)$ such that

Erdős–Falconer distance conjecture. If $|E|\geq Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then

The conjecture for the Mattila-Sjölin problem for $k$-distance sets. Assume that $C$ is a sufficiently large constant independent of $q$. Then, for every integer $k=1,2,\ldots,d$,…

Let $\mathbb{F}_q$ be a finite field, let $d\geq 2$, and let $E\subseteq\mathbb{F}_q^d$ be a $(4,s)$-Salem set. Let $\alpha(d,4,s)$ be the smallest exponent such that every $(4,s)$…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $d\geq 2$ be even, and equip $\mathbb{F}_q^d$ with the quadratic form … For $E\subseteq\mathbb{F}_q^d$, define its dista…

Falconer's distance set conjecture. If $\dim_{\mathcal{H}}E>\frac{d}{2}$, then

Let an $m$-distance set be a set of points whose pairwise distances take at most $m$ values. The multiplicity of a distance is the number of unordered pairs of points in the set re…

Let a triangular lattice be the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. Consider a regular hexagon or an equiangular hexagon w…

Let a triangular lattice be the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. An equiangular hexagon has six sides in the lattice di…

Let a triangular lattice be the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. An $m$-distance set is a set of lattice points whose p…

Let an $m$-distance set be a collection of points in the Euclidean plane such that the distances between any two points take at most $m$ possible values. A triangular lattice is th…

Let $E\subset \mathbb{R}^d$ be compact with Hausdorff dimension $\dim_{\rm H}(E)>\frac{d}{2}$. For $n\in\mathbb{N}$ and $x\in E$, let $\Delta_x^n(E)$ denote the pinned $n$-chain se…

Finite fields distance conjecture. If $\#E\geqslant Cq^{d/2}$ with $C$ sufficiently large, then

Even-dimensional monomial distance conjecture. If $\delta_{E_1,E_2}\gg p^{-n/2}$, then

Let $\mathbb{F}_q$ be a finite field of odd cardinality $q$, let $r\geq 2$, and let $E\subseteq\mathbb{F}_q^r$ have cardinality at least $cq^{\frac{r}{2}}$, where $c$ is sufficient…

Let $\mathcal{J}^{n,w}$ be the Johnson space of $w$-subsets of an $n$-element set, and let $A(\mathcal{J}^{n,w},s)$ denote the maximum size of an $s$-distance set in this space. Jo…

Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$, where $q$ is odd. For $E\subseteq\mathbb F_q^d$, define the distance set by … where … f…

Folklore conjecture. Let $C\subset\mathbb{S}^{n-1}$, with $n\geq 2$, be a spherical 4-distance 7-design. Then $C$ is a tight spherical 7-design. In particular,

Kusner's conjecture.

Bannai et al.'s classification conjecture. One of the following holds: (1) $n=2$ and $C$ is a regular hexagon or a regular heptagon; (2) $C$ is a tight spherical 5-design; or (3)…