Two nearly equal distances in $R^d$

Abstract

A point set $P \subset {\Bbb{R}}^d$ is {\it separated} if the minimum distance between any two points in $P$ is at least $1$. For $d \ne 4,5,$ we determine, for every $t_1,t_2 \ge 1$, and for $n$ at least a suitable $n_d$, the maximum number of point pairs in a separated $n$-element point set in ${\Bbb{R}}^d$, with distances in the set $[t_1,t_1 + 1]\cup[t_2,t_2 + 1]$. For $d=4,5$ we establish a weaker, similar asymptotic estimate. Recently N. Frankl and A. Kupavskii have generalized this result to unions of $k\ge 2$ intervals. We also determine the maximum number of point pairs in an $n$-element point set in ${\Bbb{R}}^d$, whose distances belong to the union of $k \ge 2$ intervals of the form $[t_i, t_i(1 + \varepsilon)]$, where $t_i > 0$ and $\varepsilon > 0$ is small.