Nearly equal distances in the plane, II

Abstract

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

9 pages