Two nearly equal distances in $R^d$
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arXiv
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| Format: | Preprint |
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2019
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| author | Erdős, P. Makai, Jr., E. Pach, J. |
| author_facet | Erdős, P. Makai, Jr., E. Pach, J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1901_01055 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Two nearly equal distances in $R^d$ Erdős, P. Makai, Jr., E. Pach, J. Metric Geometry Primary: 52C10, Secondary: 05C35 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. |
| title | Two nearly equal distances in $R^d$ |
| topic | Metric Geometry Primary: 52C10, Secondary: 05C35 |
| url | https://arxiv.org/abs/1901.01055 |