Nearly equal distances in the plane, II
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911194155057152 |
|---|---|
| author | Erdős, P. Makai, Jr., E. Pach, J. |
| author_facet | Erdős, P. Makai, Jr., E. Pach, J. |
| contents | 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 $C_{k,δ} > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_08852 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Nearly equal distances in the plane, II Erdős, P. Makai, Jr., E. Pach, J. Combinatorics Metric Geometry 52C10 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 $C_{k,δ} > 0$. |
| title | Nearly equal distances in the plane, II |
| topic | Combinatorics Metric Geometry 52C10 |
| url | https://arxiv.org/abs/2112.08852 |