On lower bounds of the density of planar periodic sets without unit distances
Alexander Tolmachev · 2024 · arXiv
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Abstract (excerpt)
Determining the maximal density $m_1(\mathbb{R}^2)$ of planar sets without unit distances is a fundamental problem in combinatorial geometry. This paper investigates lower bounds for this quantity. We introduce a novel approach to estimating $m_1(\mathbb{R}^2)$ by reformulating the problem as a Maximal Independent Set (MIS) problem on graphs constructed from flat torus, focusing on periodic sets with respect to two non-collinear vectors. Our experimental results, supported by theoretical justifications of proposed method, demonstrate that for a sufficiently wide range of parameters this approa
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Metadata source: arXiv