UC Irvine

For a compact set K ⊂ R1 and a family {Cλ}λ∈J of dynamically defined Cantor sets sufficiently close to affine with dimH K +dimH Cλ > 1 for all λ ∈ J, we prove that the sum K+Cλ 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.