On Sums of Nearly Affine Cantor Sets

For a compact set $K\subset {\mathbb{R}}^1$ and a family $\{C_{\lambda }{\}}_{\lambda \in J}$ of dynamically defined Cantor sets sufficiently close to affine with ${\text{dim}}_{H}K+{\text{dim}}_H{C}_{\lambda }>1$ for all $\lambda \in J$, under natural technical conditions we prove that the sum $K+C_{\lambda }$ has positive Lebesgue measure for almost all values of the parameter $\lambda$. 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.