On the Erdős distance problem
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866911650183905280 |
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| author | Agama, Theophilus |
| author_facet | Agama, Theophilus |
| contents | In this paper, using the compression method, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $\mathbb{R}^k$ for all $k\geq 2$, we have \begin{align} \#\bigg\{(\vec{x}_t,\vec{x_j})\in \mathbb{E}\subset\mathbb{R}^k~:~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j\leq n\bigg\}\geq C\frac{\sqrt{k}}{2}n^{1+o(1)}\nonumber \end{align} for some $C>0$. We also show that \begin{align} \# \bigg\{d_j:d_j=||\vec{x_s}-\vec{y_t}||,~d_j\neq d_i,~1\leq s,t\leq n\bigg\}\geq D\frac{\sqrt{k}}{2}n^{\frac{2}{k}-o(1)}\nonumber \end{align} for some $D>0$. These lower bounds generalize the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_00502 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Erdős distance problem Agama, Theophilus Metric Geometry Combinatorics Number Theory Primary 52C10, Secondary 51K05, 52C35 In this paper, using the compression method, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $\mathbb{R}^k$ for all $k\geq 2$, we have \begin{align} \#\bigg\{(\vec{x}_t,\vec{x_j})\in \mathbb{E}\subset\mathbb{R}^k~:~||\vec{x_j}-\vec{x_t}||=1,~1\leq t,j\leq n\bigg\}\geq C\frac{\sqrt{k}}{2}n^{1+o(1)}\nonumber \end{align} for some $C>0$. We also show that \begin{align} \# \bigg\{d_j:d_j=||\vec{x_s}-\vec{y_t}||,~d_j\neq d_i,~1\leq s,t\leq n\bigg\}\geq D\frac{\sqrt{k}}{2}n^{\frac{2}{k}-o(1)}\nonumber \end{align} for some $D>0$. These lower bounds generalize the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions. |
| title | On the Erdős distance problem |
| topic | Metric Geometry Combinatorics Number Theory Primary 52C10, Secondary 51K05, 52C35 |
| url | https://arxiv.org/abs/2002.00502 |