A new lower bound for the density of planar Sets avoiding Unit Distances
<|reference_start|>A new lower bound for the density of planar Sets avoiding Unit Distances: 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 $\delta_C = 0.22936$ for the density. To this date, no better construction with a higher lower bound has been given. In this article I give a construction planar sets with a higher density than Croft's tortoises. No explicit value for this density is given, it's just shown that Croft's density is a local minima of the density of a here constructed 1-parameter family of planar sets. So the densities are $> \delta_C$.<|reference_end|>
@article{ruhland2024a,
    title={A new lower bound for the density of planar Sets avoiding Unit Distances},
    author={Helmut Ruhland},
    journal={arXiv preprint arXiv:2408.10076},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.10076},
    primaryClass={math.MG cs.NA math.CO math.NA}
}