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 \cite{Er75f} Erd\H{o}s 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 \cite{Er97e} Erd\H{o}s 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 \cite{Er97e} Erd\H{o}s 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 \cite{KaTa04}, where\[c=\frac{48-14e}{55-16e}=0.864137\cdots.\]


References


[Er75f] Erd\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.

[Er97e] Erd\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.

[KaTa04] Katz, Nets Hawk and Tardos, G\'{a}bor, A new entropy inequality for the Erd\H{o}s distance problem. Towards a theory of geometric graphs (2004), 119-126.