Two nearly equal distances in $R^d$

Abstract

A point set $P\subset {\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$, 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 ${\mathbb{R}}^d$, whose distances belong to the union of $k\ge 2$ intervals of the form $[t_i,t_i(1+\epsilon )]$, where $t_i>0$ and $\epsilon >0$ is small.

23 pages