Algorithmic exploration of the unit distance problem in the rational plane

This paper presents reproducible experimental evidence on unit-distance graph density that surpasses recent theoretical lower bounds.

Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs.

An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation.

To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures used in the literature for generating unit-distance graphs.

Furthermore, the scaling exponent of the generated graph surpasses recent results.