  We show that under natural technical conditions, the sum of a $C^2$
dynamically defined Cantor set with a compact set in most cases (for almost all
parameters) has positive Lebesgue measure, provided that the sum of the
Hausdorff dimensions of these sets exceeds one. As an application, we show that
for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive
Lebesgue measure, while at the same time the density of states measure is
purely singular.
