Work

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

Year: 2025

Type: article

Abstract: Determining the maximal density [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]. 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.

Source: Discrete Mathematics Algorithms and Applications

DOI: 10.1142/s1793830925500314

Author: Alexander Tolmachev

Institutions: Skolkovo Institute of Science and Technology, Adyghe State University, Moscow Institute of Physics and Technology

Language: English

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W4407150837

FWCI:

Cites: 8

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Related to: 10