The behaviour of the spectral counting function for a family of sets with fractal boundaries

We construct a one-parameter family of sets in R² generated by a disjoint union of open squares. We study the spectral counting function associated to a variational Dirichlet eigenvalue problem on a set from this family and show that the spectral asymptotics depend not only on the Mmkowski dimension of the boundary, but also on whether the values of a specific function of the parameter are rational or irrational. Furthermore, we significantly sharpen the results in the rational case.