On the topology of arithmetic sums of regular Cantor sets

The study of geometrical and topological properties of arithmetic sums and differences of regular Canter sets appears naturally in distinct fields as dynamical systems (particularly, in the study of homoclinic bifurcations re lated to non-trivial hyperbolic sets) and number theory (particularly, in t he study of geometrical properties of the Markov and Lagrange spectra, rela ted to Diophantine approximations). We study the topological structure of t he sum of two regular Cantor sets and we obtain some local results related to this problem; more precisely, we give persistent examples of several dif ferent topological types of these sums or differences. AMS classification s cheme numbers: 37C45, 28A80, 28A78, 54, 26A03.