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

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Alexander Tolmachev

Published: 05 Apr 2025, Last Modified: 13 Nov 2025Discrete Mathematics, Algorithms and ApplicationsEveryoneRevisionsCC BY-SA 4.0
Abstract: Determining the maximal density m1(ℝ2) 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 m1(ℝ2) 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≤m1(ℝ2). 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.