84 problems

Falconer's distance conjecture. If $\dim_{\mathcal H}(E)>d/2$, then $\Delta(E)$ has positive Lebesgue measure.

Erdős's distinct distances conjecture. The size of $\Delta E$ is nearly linear in terms of the size of $E$. Guth and Katz settled this problem in the plane, while the formulation c…

Erdős–Falconer distance conjecture. If $|E|\gg q^m$, then $|\Delta_Q(E)|\gg q$.

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

Erdős distance conjecture for well-distributed sets. One has

Erdős–Falconer distance conjecture. If $|E|\geq Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, 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 $\mathbb{F}_q$ be a finite field of order $q$, and for $E\subset\mathbb{F}_q^d$ define … where $\\|x-y\\|=(x_1-y_1)^2+\cdots+(x_d-y_d)^2$. Assume that $q\equiv3\pmod 4$ and eit…

Let $E$ be a finite point set in $\mathbb R^2$ with cardinality $n$, and fix a distance. Erdős single-distance conjecture. No single distance can occur more than $c n^{1+\epsilon}$…

Finite-field Erdős distance conjecture. If $E$ has cardinality at least $Cq^{d/2}$, with $C$ sufficiently large, then

Erdős distance-set conjecture. If $d$ is even and $|E|\gg q^{d/2}$, then

Swanepoel's conjecture.

Even-dimensional Erdős–Falconer distance conjecture. In even dimensions, the right exponent for this threshold should be

Let $E\subset\mathbb{R}^d$, and for $x\in E$ define the pinned distance set by … Write $|\cdot|$ for one-dimensional Lebesgue measure and $\dim_{\mathcal H}(E)$ for Hausdorff dimen…

Sharp upper-bound conjecture. One should have

Let $g_d(N)$ denote the minimal number of distinct distances determined by $N$ distinct points in Euclidean space $\mathbb{R}^d$. Erdős's distinct-distances conjecture. In dimensio…

Let $E$ be a finite subset of $\mathbb R^d$, with $d e 0$ and let … be its Euclidean distance set, where $|\cdot|$ denotes the Euclidean norm and cardinality is denoted by…

Single-distance conjecture. In the special case $\partial K=S^1$, the incidence bound should improve to $q^2$, up to a slowly growing function of $q$.

Spherical-majorant conjecture. A majorant for the spherical means should suffice to prove the Erdős distance conjecture for well-distributed sets.

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 $S\subset\mathbb R^2$ be a well-distributed set, meaning that some fixed-size square contains a point of $S$, and let $K$ be a bounded convex set symmetric about the origin. De…

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…

Let $n$ be even, let $q$ be an odd prime power, and let $E\subset\mathbb{F}_q^n$. Define … and define the distance set … Let $\alpha_n$ be the smallest exponent such that…

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