Erdős's unit distance problem and rigidity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915402853908480 |
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| author | Pach, János Raz, Orit E. Solymosi, József |
| author_facet | Pach, János Raz, Orit E. Solymosi, József |
| contents | According to a classical result of Spencer, Szemerédi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erdős's lower bound, $n^{1+O(1/\log\log n)}$, which is conjectured to be optimal. We prove a structural result for point sets with nearly $n^{4/3}$ unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by the last two authors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Erdős's unit distance problem and rigidity Pach, János Raz, Orit E. Solymosi, József Combinatorics According to a classical result of Spencer, Szemerédi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erdős's lower bound, $n^{1+O(1/\log\log n)}$, which is conjectured to be optimal. We prove a structural result for point sets with nearly $n^{4/3}$ unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by the last two authors. |
| title | Erdős's unit distance problem and rigidity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.15679 |