Favourite distances in high dimensions

Let $S$ be a set of $n$ points in $d$-dimensional Euclidean space. Assign to each $x\in S$ an arbitrary distance $r(x)>0$. Let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erdös and Pach (1988) introduced the extremal quantity $f_d(n)=\max\sum_{x\in S}e_r(x,S)$, where the maximum is taken over all $n$-point sets $S$ in $d$-dimensional space and all assignments $r\colon S\to(0,\infty)$ of distances. We give a quick derivation of the asymptotics of the error term of $f_d(n)$ using only the analogous asymptotics of the maximum number of unit distance pairs in a set of $n$ points, which improves on previous results of Avis, Erdös and Pach (1988) and Erdös and Pach (1990). Then we prove a stability result for $d\geq 4$, asserting that if $(S,r)$ with $|S|=n$ satisfies $e_r(S)=f_d(n)-o(n^2)$, then, up to $o(n)$ points, $S$ is a Lenz construction with $r$ constant. Finally we use stability to show that for $n$ sufficiently large (depending on $d$) the pairs $(S,r)$ that attain $f_d(n)$ are up to scaling exactly the Lenz constructions that maximise the number of unit distance pairs with $r\equiv 1$, with some exceptions in dimension 4. Analogous results hold for the furthest neighbour digraph, where $r$ is fixed to be $r(x)=\max_{y\in S} |xy|$ for $x\in S$.