Nearly equal distances in the plane, II

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 $\delta >0$. Suppose for all $1\le \mathrm{\ell }(1)\le \mathrm{\ell }(2)<\mathrm{\ell }(3)\le k$ that $\mathrm{\mid }{t}_{\mathrm{\ell }(3)}\mathrm{/}(t_{\mathrm{\ell }(1)}+t_{\mathrm{\ell }(2)})-1\mathrm{\mid }\ge \delta$. Then for $n\ge n_{k,\delta }$, the number of pairs $\{p_i,p_j\}$, for which $d(p_i,p_j)\in [t_1,t_1+1]\cup \dots \cup [t_k,t_k+1]$, is at most $n^2\mathrm{/}4+C_{k,\delta }n$. This is sharp, up to the value of the constant $C_{k,\delta }>0$.