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| Автор: | |
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| Формат: | Preprint |
| Опубліковано: |
2023
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| Предмети: | |
| Онлайн доступ: | https://arxiv.org/abs/2310.06821 |
| Теги: |
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| _version_ | 1866915335563640832 |
|---|---|
| author | Zakharov, Dmitrii |
| author_facet | Zakharov, Dmitrii |
| contents | We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any measurable set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the constant $c_0$.
Moreover, we show that for all $2\le k \le n$ a set $A$ avoiding $k$ pairwise orthogonal vectors has measure at most $\exp(-c_1 \min\{\sqrt{n}, n/k\})$ for some $c_1>0$. Proofs rely on the harmonic analysis on the sphere and the hypercontractive inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06821 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spherical sets avoiding orthonormal bases Zakharov, Dmitrii Metric Geometry Combinatorics We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any measurable set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the constant $c_0$. Moreover, we show that for all $2\le k \le n$ a set $A$ avoiding $k$ pairwise orthogonal vectors has measure at most $\exp(-c_1 \min\{\sqrt{n}, n/k\})$ for some $c_1>0$. Proofs rely on the harmonic analysis on the sphere and the hypercontractive inequality. |
| title | Spherical sets avoiding orthonormal bases |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2310.06821 |