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

2024/08/19 by Ruhland, Helmut
#52C17 52C10 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2408.10076

Abstract

In a recently published article by G. Ambrus et al. a new 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 lower bound δ_C = 0.22936 for the density. To this date, no better construction with a higher bound has been given. In the 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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