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

TL;DR

This paper explores lower bounds for the maximum density of planar sets without unit distances, introducing a new graph-based approach and comparing various algorithms, but finds limited improvements over existing bounds.

Contribution

It presents a novel reformulation of the problem as a Maximal Independent Set problem on torus-based graphs and evaluates multiple algorithms for this task.

Findings

The new approach does not significantly improve the known lower bound.

Best sets found approximate Croft's construction.

Comparison of open source MIS algorithms on this problem.

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.