Two nearly equal distances in $R^d$

TL;DR

This paper determines the maximum number of point pairs with distances in specific unions of intervals in high-dimensional Euclidean spaces, extending previous results to multiple intervals and nearly equal distances.

Contribution

It generalizes existing results by establishing maximum pair counts for separated point sets with distances in unions of multiple intervals in ${ m R}^d$, including near-equal distances.

Findings

Maximum pairs for two intervals in ${ m R}^d$ for $d e 4,5$

Asymptotic estimates for $d=4,5$

Extension to unions of $k extgreater 2$ intervals with small perturbations

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.