The study of geometrical and topological properties ofarithmeticsums and differences of regular Cantor sets appears naturally indistinct fields as dynamical systems (particularly, in the study ofhomoclinic bifurcations related to non-trivial hyperbolic sets) and numbertheory (particularly, in the study of geometrical properties of theMarkovand Lagrange spectra, related to Diophantine approximations). We studythe topological structure of the sum of two regularCantor sets and we obtain some local results related to thisproblem; more precisely, we give persistent examples of severaldifferent topological types of these sums or differences.