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On the topology of arithmetic sums of regular Cantor sets

The study of geometrical and topological properties of arithmetic sums and differences of regular Cantor sets appears naturally in distinct fields as dynamical systems (particularly, in the study of homoclinic bifurcations related to non-trivial hyperbolic sets) and number theory (particularly, in the study of geometrical properties of the Markov and Lagrange spectra, related to Diophantine approximations). We study the topological structure of the sum of two regular Cantor sets and we obtain some local results related to this problem; more precisely, we give persistent examples of several different topological types of these sums or differences.