Sums of two homogeneous Cantor sets

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture. We also consider self-similar sets with overlaps on the real line (not necessarily homogeneous), and show that one can create an interval by applying arbitrary small perturbations, if the uniform self-similar measure has $L^2$-density.