No new lower bound for the density of planar Sets avoiding Unit Distances
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918062487240704 |
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| author | Ruhland, Helmut |
| author_facet | Ruhland, Helmut |
| contents | In a recently published article by G. Ambrus et al. a new \emph{upper bound} for the density of an unit avoiding, periodic set is given as $0.2470$, the first upper bound $< 1/4$. A construction of Croft 1967 gave a \emph{lower bound} $δ_C = 0.22936$ for the density. To this date, no better construction with a higher bound has been given. In the \emph{first versions} of this article I gave a construction of planar sets with a "higher" density than Croft's tortoises. No explicit value for this density was given, it was just shown that Croft's density is a local minima in the density of the constructed 1-parameter family of planar sets. But now I found a servere error. After the correction in this article none of the investigated sets of constant diameter resulted in a new lower bound. I did not withdraw the article, maybe something could be useful for somebody. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | No new lower bound for the density of planar Sets avoiding Unit Distances Ruhland, Helmut Metric Geometry Numerical Analysis Combinatorics 52C17 52C10 In a recently published article by G. Ambrus et al. a new \emph{upper bound} for the density of an unit avoiding, periodic set is given as $0.2470$, the first upper bound $< 1/4$. A construction of Croft 1967 gave a \emph{lower bound} $δ_C = 0.22936$ for the density. To this date, no better construction with a higher bound has been given. In the \emph{first versions} of this article I gave a construction of planar sets with a "higher" density than Croft's tortoises. No explicit value for this density was given, it was just shown that Croft's density is a local minima in the density of the constructed 1-parameter family of planar sets. But now I found a servere error. After the correction in this article none of the investigated sets of constant diameter resulted in a new lower bound. I did not withdraw the article, maybe something could be useful for somebody. |
| title | No new lower bound for the density of planar Sets avoiding Unit Distances |
| topic | Metric Geometry Numerical Analysis Combinatorics 52C17 52C10 |
| url | https://arxiv.org/abs/2408.10076 |