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arXiv · 2408.10076

No new lower bound for the density of planar Sets avoiding Unit Distances

Helmut Ruhland
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Abstract

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.

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Helmut Ruhland. 2025-06-18. No new lower bound for the density of planar Sets avoiding Unit Distances. https://arxiv.org/abs/2408.10076

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