Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Can there be $\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?

Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Can there be $\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?

The answer is yes: Bhowmick [Bh24] constructs a set of $n$ points in $\mathbb{R}^2$ such that $\lfloor\frac{n}{4}\rfloor$ distances occur at least $n+1$ times.