Nearly equal distances in the plane, II

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 $C_{k,\delta } > 0$.