For a compact set K ⊂ R 1 and a family {C λ } λ∈J of dynamically defined Cantor sets sufficiently close to affine with dimH K + dimH C λ > 1 for all λ ∈ J, under natural technical conditions we prove that the sum K +C λ has positive Lebesgue measure for almost all values of the parameter λ. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.