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

TL;DR

This paper investigates bounds on the density of planar sets avoiding unit distances, showing that previous constructions do not improve the known lower bound after correction of an earlier error.

Contribution

The paper clarifies that previous attempts to improve the lower bound for the density of such sets are invalidated by a correction, reaffirming Croft's lower bound.

Findings

No new lower bound was found after correction.

Previous constructions do not surpass Croft's density.

The correction impacts earlier claims of improved bounds.

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} $\delta_C = 0.22936$ 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.