Nearly equal distances in the plane, II

TL;DR

This paper investigates the distribution of distances among separated points in the plane, establishing bounds on the number of pairs with distances in specified intervals under certain ratio conditions.

Contribution

It provides a new upper bound on the number of point pairs with distances in specified ranges, extending understanding of distance distributions in separated point sets.

Findings

Bound on pairs with distances in specified intervals is at most n^2/4 + C_{k,δ}n.

Result is sharp up to a constant factor.

Conditions on ratios of interval endpoints are crucial for the bound.

Abstract

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1 \le \ell (1) \le \ell (2) < \ell (3) \le k$ that $|t_{\ell (3)} / (t_{\ell (1)} + t_{\ell (2)}) - 1| \ge \delta $. Then for $n \ge n_{k, \delta }$, the number of pairs $\{ p_i,p_j\} $, for which $d(p_i,p_j) \in [t_1, t_1 + 1] \cup \ldots \cup [t_k, t_k + 1] $, is at most $n^2/4 + C_{k,\delta }n$. This is sharp, up to the value of the constant $C_{k,\delta } > 0$.