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arXiv · 2411.13248

On lower bounds of the density of planar periodic sets without unit distances

Alexander Tolmachev
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Abstract

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 approach does not improve the known lower bound $0.22936 \le m_1(\mathbb{R}^2)$. The best discrete sets found are approximations of Croft's construction. In addition, several open source software packages for MIS problem are compared on this task.

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Alexander Tolmachev. 2025-04-11. On lower bounds of the density of planar periodic sets without unit distances. https://doi.org/10.1142/s1793830925500314

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