Obtuse almost-equiangular sets
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913802289676288 |
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| author | Bachoc, Christine Bekker, Bram Moustrou, Philippe Filho, Fernando Mário de Oliveira |
| author_facet | Bachoc, Christine Bekker, Bram Moustrou, Philippe Filho, Fernando Mário de Oliveira |
| contents | For $t \in [-1, 1)$, a set of points on the $(n-1)$-dimensional unit sphere is called $t$-almost equiangular if among any three distinct points there is a pair with inner product $t$. We propose a semidefinite programming upper bound for the maximum cardinality $α(n, t)$ of such a set based on an extension of the Lovász theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of $n$ and $t$ it is better.
We also refine existing spectral methods to show that $α(n, t) \leq 2(n+1)$ for all $n$ and $t \leq 0$, with equality only at $t = -1/n$. This allows us to show the uniqueness of the optimal construction at $t = -1/n$ for $n \leq 5$ and to enumerate all possible constructions for $n \leq 3$ and $t \leq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11086 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Obtuse almost-equiangular sets Bachoc, Christine Bekker, Bram Moustrou, Philippe Filho, Fernando Mário de Oliveira Combinatorics Metric Geometry Optimization and Control 52C10, 51K99, 90C22 For $t \in [-1, 1)$, a set of points on the $(n-1)$-dimensional unit sphere is called $t$-almost equiangular if among any three distinct points there is a pair with inner product $t$. We propose a semidefinite programming upper bound for the maximum cardinality $α(n, t)$ of such a set based on an extension of the Lovász theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of $n$ and $t$ it is better. We also refine existing spectral methods to show that $α(n, t) \leq 2(n+1)$ for all $n$ and $t \leq 0$, with equality only at $t = -1/n$. This allows us to show the uniqueness of the optimal construction at $t = -1/n$ for $n \leq 5$ and to enumerate all possible constructions for $n \leq 3$ and $t \leq 0$. |
| title | Obtuse almost-equiangular sets |
| topic | Combinatorics Metric Geometry Optimization and Control 52C10, 51K99, 90C22 |
| url | https://arxiv.org/abs/2504.11086 |