Problem Statement
Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or even $\gg n/\sqrt{\log n}$?
Categories:
Geometry Distances
Progress
The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\sqrt{\log n}$ would be best possible.It may be true that there are $\gg n$ many such points, or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in [Er75f] Erdős conjectured\[\sum_{x\in A}d(x) \gg \frac{n^2}{\sqrt{\log n}},\]where $A\subset \mathbb{R}^2$ is any set of $n$ points.
In [Er97e] Erdős offers \$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\gg n$ many such points.
In [Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.
The best known bound is\[\gg n^{c-o(1)},\]due to Katz and Tardos [KaTa04], where\[c=\frac{48-14e}{55-16e}=0.864137\cdots.\]
Source: erdosproblems.com/604 | Last verified: January 15, 2026