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

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.