  We construct "large" Cantor sets whose complements resemble the unit disk
arbitrarily well from the point of view of the squeezing function, and we
construct "large" Cantor sets whose complements do not resemble the unit disk
from the point of view of the squeezing function. Finally we show that
complements of Cantor sets arising as Julia sets of quadratic polynomials have
degenerate squeezing functions, despite of having Hausdorff dimension
arbitrarily close to two.
