Adapting the Aloupis–Hearn–Iwasawa–Uehara method to the parametric setting

Develop an adaptation of the Aloupis–Hearn–Iwasawa–Uehara vertical-line slice analysis (used to prove the lower bound σ₂ ≥ 12 for disjoint unit-disk coverings) to the parameterized family 𝒜₂^ρ of overlapping unit-disk coverings employed in the paper’s function f(ρ, k), so that the method can be beneficially applied in the parametric context.

Background

The authors obtain lower bounds on exact covering via two approaches: (i) combining the Aloupis–Hearn–Iwasawa–Uehara (AHIU) lower bound σ₂ ≥ 12 with an extension argument to handle boundary points, and (ii) a new parametric method that varies the overlap parameter ρ in a hexagonal-lattice-based covering and optimizes a derived function f(ρ, k).

They note that AHIU’s vertical-line slice analysis was not used within the parametric method, as they could not adapt it in a way that improved the parametric bounds. A successful adaptation could potentially strengthen the parametric lower bounds or unify the two techniques.