P. Erdős | Searcharxiv

arXiv2025

Nearly equal distances in the plane, II

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 $δ> 0$. Suppose for all $1 \le \ell (1) \le \ell (2) < \ell (3) \le k$ that $|t_{\ell (3)} / (t_{\ell (1)} + t_{\ell (2)}) - 1| \ge δ$. Then for $n \ge n_{k, δ}$, 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 $n^2/4 + C_{k,δ}n$. This is sharp, up to the value of the constant $C_{k,δ} > 0$.

math.CO↗

arXiv2025

Two nearly equal distances in $R^d$

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.

math.MG↗